Lab Section

In this lab, we will go over regularization, classification and performance metrics. We will be using the caret package in R. https://topepo.github.io/caret/train-models-by-tag.html

Perfomance Metrics

K- fold cross validatation - Resampling method

Randomly split the training data into k folds. If you specify 10 folds, then you split the data into 10 partitions. Train the model on 9 of those partitions, and test your model on the 10th partition. Iterate through until every partition has been held out.

A smaller k is more biased, but a larger k can be very variable.

Bootstrapping - Resampling method

Sample with replacement. Some samples may be represented several times within the boostrap sample, while others may not be represented at all. The samples that are not selected are called out of bag samples.

Boostrap error rates usually have less uncertainty than k-fold cross validation, but higher bias.

Error

Deviation of the observed value to the true value (population mean)

Residual

Deviation of the observed value to the estimated value (sample mean) \[residual=y_i - \hat{y_i}\] where \(\hat{y_i}\) is the estimated value

Mean Squared Error (MSE)

\[MSE=\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y_i})^2\]

Root Mean Squared Error (RMSE)

Same units as original data.

\[RMSE=\sqrt{MSE}\]

R^2

Proportion of information explained by the model. It is a measure of correlation, not accuracy. \[1-RSS/TSS\]

L2 regularization : Ridge regression. Regularize by adding the sum of the coefficients, squared, to the function.

\[Ridge Regression=\sum_{i=1}^{n}(y_i - w_0 - \sum_{j=1}^{p}w_jx_{ij})^2 + \lambda\sum_{j=1}^p(w_j)^2\]

L1 regularization : Lasso Regression. Regularize by adding the sum of the absolute value of the coefficients to the model. Coefficient estimates may be pushed to zero – Lasso can perform variable selection

\[Lasso Regression=\sum_{i=1}^{n}(y_i - w_0 - \sum_{j=1}^{p}w_jx_{ij})^2 + \lambda\sum_{j=1}^p|w_j|\]

Sensitivity or True Positive Rate

TP = True Positives TN = True Negatives FP = False Positives - Type I error FN = False Negatives - Type II error N = actual negative samples P = actual positive samples

\[TPR=TP/(TP + FN)\]

Specificity or True Negative Rate

\[TNR=TN/(TN + FP)\]

Receiver Operating Characteristics (ROC)

Plot of True Positive Rate (sensitivity) against False Positive Rate, or plots the True Positive Rate (sensitivity) against specificity.

Either way, a good ROC curves up through the left corner, and has a large area underneath.

Area under ROC curve (AUC)

The area underneath the ROC curve

Logistic function:

\[P(X)=e^{w_0 + w_1X}/{1+e^{w_0+w_1X}}\]

The broad steps of Machine learning in R.

  1. Split the data into training and test. Set test aside.

  2. Fit a good model to the training data. This includes using bootstapping, cross validation etc. to resample the training data and fit a good model.

  3. Visualize if your model learned on the training data by looking at ROC curve and AUC.

  4. Test how your model performs on the test data.

Broad steps for choosing between models according to Max Kuhn and Kjell Johnson

  1. Start with several models that are the least interpretable and the most flexible, like boosted trees and svms. These models are the often the most accurate.

  2. Investigate simpler models that are less opaque, like partial least squares, generalized additive models, or naive bayes models.

  3. Consider using the simplest model that reasonable approximates the performance of more complex models

head(airquality)
##   Ozone Solar.R Wind Temp Month Day
## 1    41     190  7.4   67     5   1
## 2    36     118  8.0   72     5   2
## 3    12     149 12.6   74     5   3
## 4    18     313 11.5   62     5   4
## 5    NA      NA 14.3   56     5   5
## 6    28      NA 14.9   66     5   6

Split data into training and test set

train_size <- floor(0.75 * nrow(airquality))
set.seed(543)
train_pos <- sample(seq_len(nrow(airquality)), size = train_size)
train_regression <- airquality[train_pos,-c(1,2)]#remove first two columns
test_regression <- airquality[-train_pos,-c(1,2)]

dim(train_regression) 
## [1] 114   4
dim(test_regression)
## [1] 39  4
#train_regression
dim(train_regression)
## [1] 114   4

Resampling in R

(method = boot, cv, LOOCV …)

?trainControl

Ridge Regression

\[Ridge Regression=\sum_{i=1}^{n}(y_i - w_0 - \sum_{j=1}^{p}w_jx_{ij})^2 + \lambda\sum_{j=1}^p(w_j)^2\] 2. Create and train model

ctrl =  trainControl(method = "boot", 15)
# bootstrapped 15 repititions

Ridge_regression <- train(Temp ~ Wind + Month, data = train_regression,
                          method = 'ridge', trControl= ctrl) 
Ridge_regression 
## Ridge Regression 
## 
## 114 samples
##   2 predictor
## 
## No pre-processing
## Resampling: Bootstrapped (15 reps) 
## Summary of sample sizes: 114, 114, 114, 114, 114, 114, ... 
## Resampling results across tuning parameters:
## 
##   lambda  RMSE      Rsquared   MAE     
##   0e+00   8.185003  0.2868549  6.716834
##   1e-04   8.185000  0.2868560  6.716832
##   1e-01   8.182914  0.2877921  6.715122
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was lambda = 0.1.

Examine the residuals

ridge_test_pred <- predict(Ridge_regression, newdata = test_regression)

# plot the predicted values vs the observed values
plot_ridge_test_pred <- data.frame(Temp_test_pred = ridge_test_pred, 
                                   Observed_Temp = test_regression$Temp)
#dim(plot_ridge_test_pred)
ggplot(data = plot_ridge_test_pred) +
  geom_point(aes(x=Observed_Temp, y = Temp_test_pred)) + 
  ggtitle("True Temp Value vs Predicted Temp Value Ridge Regression") +
  theme_bw()

# median residual value should be close to zero
median(resid(Ridge_regression))
## [1] 0.2348498

Homework

Lasso

\[Lasso Regression=\sum_{i=1}^{n}(y_i - w_0 - \sum_{j=1}^{p}w_jx_{ij})^2 + \lambda\sum_{j=1}^p|w_j|\] 2. Create and train model

ctrl =  trainControl(method = "boot", 9)
# bootstrapped 15 repititions

Lasso_regression <- train(Temp ~ Wind + Month, data = train_regression,
                          method = 'lasso', trControl= ctrl)
Lasso_regression
## The lasso 
## 
## 114 samples
##   2 predictor
## 
## No pre-processing
## Resampling: Bootstrapped (9 reps) 
## Summary of sample sizes: 114, 114, 114, 114, 114, 114, ... 
## Resampling results across tuning parameters:
## 
##   fraction  RMSE      Rsquared   MAE     
##   0.1       8.832033  0.1630620  7.093072
##   0.5       8.242107  0.1968353  6.664605
##   0.9       8.151890  0.2065470  6.556639
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was fraction = 0.9.

Examine the residuals (on test):

lasso_test_pred <- predict(Lasso_regression, newdata = test_regression)

# plot the predicted values vs the observed values
plot_lasso_test_pred <- data.frame(Temp_test_pred_lasso = lasso_test_pred, 
                                   Observed_Temp = test_regression$Temp)
#dim(plot_lasso_test_pred)
ggplot(data = plot_lasso_test_pred) +
  geom_point(aes(x=Observed_Temp, y = Temp_test_pred_lasso)) + 
  ggtitle("True Temp Value vs Predicted Temp Value Lasso Regression") +
  theme_bw()

# median residual value should be close to zero
median(resid(Lasso_regression))
## [1] 0.07194436

Classification

  1. Split into training and test set
data(iris)

#split into training and test set 
train_size <- floor(0.75 * nrow(iris))
set.seed(543)
train_pos <- sample(seq_len(nrow(iris)), size = train_size)
train_pos
##   [1] 103  99 113 132 112 104  94 143  60 147  76   5  59  92 111 128  86
##  [18] 123  36  77  93   9  21  25  13 138  56 129  87  54 105  65  52 145
##  [35] 144 110  95  38  88  72  89  18 140   6  42   7 127  83  14  69 130
##  [52]  71  39  20  48 109  51 120  23  97 119  15  27  47 133  17  75  64
##  [69] 100  19 117  98 106  40  62 114  43 135 149 150 108  37  67  32 146
##  [86] 122  91   3  11  73  96 115  33  22  29  84  30  68  61  55 118 121
## [103] 148  24  10  50  57  53  80  44  34  74
#???vector with 117 indexes of randomly sampled train set observations
train_classifier <- iris[train_pos,]
test_classifier <- iris[-train_pos,]

dim(train_classifier) #112x5
## [1] 112   5
dim(test_classifier) #38x5
## [1] 38  5

Linear Discriminant analysis

  • Good for well separated classes,
  • more stable with small n than logistic regression,
  • and good for more than 2 response classes.
  • LDA assumes a normal distribution with a class specific mean and common variance.

Let’s see if our (unsplitted) data follows the assumptions of LDA.

slength <- ggplot(data = iris, aes(x = Sepal.Length, fill = Species)) + 
  geom_histogram(position="identity", alpha=0.5, bins= 25)  +
  ggtitle("Sepal Length by iris species")+
  theme_bw()
slength

swidth <- ggplot(data = iris, aes(x = Sepal.Width, fill = Species)) + 
  geom_histogram(position="identity", alpha=0.5, bins= 25) +
  ggtitle("Sepal width by iris species")+
  theme_bw()
swidth

plength <- ggplot(data = iris, aes(x = Petal.Length, fill = Species)) + 
  geom_histogram(position="identity", alpha=0.5, bins= 25) +
  ggtitle("Petal lenth by iris species")+
  theme_bw()
plength

pwidth <- ggplot(data = iris, aes(x = Petal.Width, fill = Species)) + 
  geom_histogram(position="identity", alpha=0.5, bins= 25) +
  ggtitle("Petal width by iris species")+
  theme_bw()
pwidth

grid.arrange(slength, swidth, plength, pwidth)

Note Conclusion on LDA suitability check with Xj histograms: n is rather small each level of response distribution is close to normal more than 2 levels of response Y well separated classes?..

LDA analysis on train set:

Note: lda(y~ x1+x2…, data, cv=T)

LDA <- lda(Species~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width, 
           data= train_classifier, cv= T)
LDA
## Call:
## lda(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width, 
##     data = train_classifier, cv = T)
## 
## Prior probabilities of groups:
##     setosa versicolor  virginica 
##  0.3214286  0.3571429  0.3214286 
## 
## Group means:
##            Sepal.Length Sepal.Width Petal.Length Petal.Width
## setosa         5.002778    3.441667     1.441667   0.2472222
## versicolor     5.975000    2.802500     4.297500   1.3425000
## virginica      6.625000    2.961111     5.602778   2.0166667
## 
## Coefficients of linear discriminants:
##                     LD1        LD2
## Sepal.Length  0.8688807  0.0250930
## Sepal.Width   1.3090031  2.2033404
## Petal.Length -2.1733913 -0.9306213
## Petal.Width  -2.9542135  2.9217844
## 
## Proportion of trace:
##    LD1    LD2 
## 0.9925 0.0075

Note Conclusion of LDA: * ‘LDA probabilities of groups’ shows ~equally represented class/response levels * ‘Group means’ ?????? Xj not really normally distributed * ‘Proportion of trace’ - percentage separation achieved by each discriminant function.Number of discriminant functions=(j-1), where j is a number levels in the response. Therefore, in LDA j>2.

!LD1 function is the best to explain the data variance. Sepal.length and Sepal.width are the two variables with positive strong relationship with this function.

  1. Test model on test set
#predict the species of the test data
LDA_predict <- predict(LDA, newdata=test_classifier)
LDA_predict
## $class
##  [1] setosa     setosa     setosa     setosa     setosa     setosa    
##  [7] setosa     setosa     setosa     setosa     setosa     setosa    
## [13] setosa     setosa     versicolor versicolor versicolor versicolor
## [19] versicolor versicolor versicolor versicolor versicolor versicolor
## [25] virginica  virginica  virginica  virginica  virginica  virginica 
## [31] virginica  virginica  versicolor virginica  virginica  virginica 
## [37] virginica  virginica 
## Levels: setosa versicolor virginica
## 
## $posterior
##           setosa   versicolor    virginica
## 1   1.000000e+00 8.927588e-22 1.502714e-42
## 2   1.000000e+00 5.608958e-18 7.859057e-38
## 4   1.000000e+00 1.340664e-16 8.780825e-36
## 8   1.000000e+00 7.063790e-20 4.874210e-40
## 12  1.000000e+00 3.071910e-18 9.278990e-38
## 16  1.000000e+00 2.450746e-26 1.808054e-47
## 26  1.000000e+00 1.741777e-16 8.142385e-36
## 28  1.000000e+00 3.298574e-21 8.515021e-42
## 31  1.000000e+00 2.177639e-16 1.541993e-35
## 35  1.000000e+00 1.139046e-17 2.613655e-37
## 41  1.000000e+00 3.067677e-21 1.548311e-41
## 45  1.000000e+00 8.513016e-17 6.298481e-35
## 46  1.000000e+00 1.619854e-16 1.480553e-35
## 49  1.000000e+00 8.466069e-23 8.730428e-44
## 58  1.428152e-13 1.000000e+00 3.990264e-08
## 63  2.374740e-17 9.999997e-01 3.470145e-07
## 66  2.056260e-17 9.999767e-01 2.334426e-05
## 70  4.162354e-17 9.999987e-01 1.266945e-06
## 78  5.219725e-27 7.577590e-01 2.422410e-01
## 79  3.553343e-23 9.952595e-01 4.740522e-03
## 81  3.715969e-17 9.999989e-01 1.097426e-06
## 82  3.965139e-15 9.999999e-01 1.096980e-07
## 85  1.036329e-24 9.716164e-01 2.838356e-02
## 90  1.350592e-20 9.999173e-01 8.267215e-05
## 101 5.859828e-53 3.814340e-09 1.000000e+00
## 102 5.479761e-38 1.245695e-03 9.987543e-01
## 107 5.707472e-33 6.378269e-02 9.362173e-01
## 116 8.008099e-41 1.986249e-05 9.999801e-01
## 124 3.240609e-31 1.348781e-01 8.651219e-01
## 125 4.511111e-40 7.110302e-05 9.999289e-01
## 126 9.318183e-37 2.895450e-03 9.971045e-01
## 131 9.974593e-42 1.875211e-04 9.998125e-01
## 134 2.332648e-28 8.097276e-01 1.902724e-01
## 136 8.719380e-46 2.112584e-06 9.999979e-01
## 137 2.102117e-45 5.219049e-07 9.999995e-01
## 139 2.649695e-29 2.178681e-01 7.821319e-01
## 141 1.371274e-45 9.014290e-07 9.999991e-01
## 142 1.702640e-36 4.062736e-04 9.995937e-01
## 
## $x
##             LD1         LD2
## 1    8.09659797  0.24247894
## 2    7.26832029 -0.86420986
## 4    6.92121725 -0.74446585
## 8    7.66147047 -0.07342653
## 12   7.27035519 -0.17150726
## 16   8.98784735  2.73183587
## 26   6.92053011 -1.04782483
## 28   7.96614692  0.15192611
## 31   6.87765427 -0.83250938
## 35   7.18188147 -0.73693795
## 41   7.93162767  0.62521022
## 45   6.81176056  1.02252730
## 46   6.88601087 -0.57454072
## 49   8.31483561  0.59510349
## 58  -0.00989576 -1.61696704
## 63  -0.83730146 -2.68146774
## 66  -1.10202409  0.11556898
## 70  -0.87023505 -1.64526224
## 78  -3.42322321  0.21339749
## 79  -2.48480169 -0.14354788
## 81  -0.87068430 -1.77504345
## 82  -0.35792383 -1.97415976
## 85  -2.87522982  0.06173036
## 90  -1.76530495 -1.15647678
## 101 -7.91483665  2.27116866
## 102 -5.40609861  0.02110643
## 107 -4.55501442 -0.46812946
## 116 -5.84663229  2.12042195
## 124 -4.24155864 -0.07240126
## 125 -5.73358157  1.39167848
## 126 -5.19579484  0.02816921
## 131 -6.05838040 -0.64903204
## 134 -3.65907254 -0.91472681
## 136 -6.71760097  0.96787771
## 137 -6.61915848  2.57157278
## 139 -3.89218282  0.67413510
## 141 -6.66430711  1.92060786
## 142 -5.10841398  2.09875866

Notes (https://www.displayr.com/linear-discriminant-analysis-in-r-an-introduction/):

LDA OUTPUT FOR VALIDATION SET:

$class - shows actual class level (in validation set)

$posterior - is the regressional weight for each indexed observation in validation set. The highest score predicts the sub-class.

ONE CAN COMPARE class=actual vs. posterior=predicted (in valid.set)

*2 LD functions are run parallely for each n. Mean LD1 and LD2 (for each n) is calculated next and the best LD function is selected (‘LD trace’. Tracing that LD one can see the features strongly correlated with it) .

3(=Yj) mean d.scores are collected for each feature (Xi). It will be a df [j:i]. The highest disc score will reveal the ‘j’ sub-class prediction. R^2 correlation coefficient is calculated for this prediction. All R^2 scores are collected (one for each feature, a vector with i length). The highest R^2 score identifies the discriminating attribute.

Thus, LDA is not only a supervised classifing method but also a dimension reduction technique as well (for not large n sample with independent and normally distr-ed Xi and Yj>2).

$x LD1 and LD2 -linear discriminants (remember that in this case with 3 classes we have at most two linear discriminants) are combination coefficients for each of indexed observation in validation set.

**** $singular values (svd) - gives the ratio of the between- and within-group standard deviations on the linear discriminant variables (LDs). {not shown in this output} ***** $prop -We can use the singular values to compute the amount of the between-group variance that is explained by each linear discriminant. In our example we see that the first linear discriminant explains more than {99%} of the between-group variance in the iris dataset. {not shown in this output}

Confusion Matrix A confusion matrix is a technique for summarizing the performance of a classification algorithm. Classification accuracy alone can be misleading if you have an unequal number of observations in each class or if you have more than two classes in your dataset.

#?confusionMatrix()
#install.packages("e1071")
confusionMatrix(LDA_predict$class, reference = test_classifier$Species)
## Confusion Matrix and Statistics
## 
##             Reference
## Prediction   setosa versicolor virginica
##   setosa         14          0         0
##   versicolor      0         10         1
##   virginica       0          0        13
## 
## Overall Statistics
##                                           
##                Accuracy : 0.9737          
##                  95% CI : (0.8619, 0.9993)
##     No Information Rate : 0.3684          
##     P-Value [Acc > NIR] : 2.196e-15       
##                                           
##                   Kappa : 0.9603          
##                                           
##  Mcnemar's Test P-Value : NA              
## 
## Statistics by Class:
## 
##                      Class: setosa Class: versicolor Class: virginica
## Sensitivity                 1.0000            1.0000           0.9286
## Specificity                 1.0000            0.9643           1.0000
## Pos Pred Value              1.0000            0.9091           1.0000
## Neg Pred Value              1.0000            1.0000           0.9600
## Prevalence                  0.3684            0.2632           0.3684
## Detection Rate              0.3684            0.2632           0.3421
## Detection Prevalence        0.3684            0.2895           0.3421
## Balanced Accuracy           1.0000            0.9821           0.9643
  • kappa - or Cohen’s coefficient is similar to correlation coefficients, it can range from −1 to +1, where 0 represents the amount of agreement that can be expected from random chance, and 1 represents perfect agreement between the raters. While kappa values below 0 are possible, Cohen notes they are unlikely in practice
#Activation function maps Pred.score [-int:+int] to Probability (0 of PS mamps to 0.5 of Prob. Max of PS maps to max of Prob.=1, same for the mins.) 

# LDA_predict$posterior = df[idx:PredScore for each 3Species] = df[38:3], where index is not incl in the dataframe

# save the predictions in a new variable
predictions <- as.data.frame(LDA_predict$posterior) %>% 
  rownames_to_column("idx") # rownames (indexes of validation set) are now officialy under column name 'idx' and are included in df. The df is now one column bigger = [38:4] 

test_classifier <- test_classifier %>% 
  rownames_to_column("idx")
#now test_classifier has officially Index as a row name.
#df[38:5] became [38:6]

predictions_actual <- full_join(predictions,test_classifier, by = "idx" ) #join and order by Index
#predictions =df[38:4]. Idx;regression weight for 3 sub-classesY1-3
#predictions_actual=df[38:9]. Idx,predicted RWeights for all Y1-3, actual X1-4, actual_Y class


# choose the two classes we want to compare, setosa and versicolor
set_vers_true_labels <- predictions_actual %>% 
  filter(Species %in% c("setosa", "versicolor")) %>% 
  #[24:9] filtered 24 obs where Y1 or Y2 have the discriminating PredScore (all observations with high PredSc for Y3 is not here)
  mutate(Species = as.character(Species)) 
  #(df[24:9]; idx, PredScore for Y1-3 (where Y3 score is never discr-ve), actual X1-4, actual_class for these 24 predictions)

#make dataframe PredScores for setosa and the actual labels
pred_label <- data.frame(prediction = set_vers_true_labels$setosa,
                         label = set_vers_true_labels$Species)
#df[24:2]

# summary of prediction:
ggplot(pred_label, aes(x = 1:24, y = prediction, color = label))+
  geom_point()+
  ggtitle("Prediction Scores calculated for Setosa in test set are mapped to\n probabilities using activating function. As a result,in test set, 14\n observations were\n predicted by LDA as setosa and 10 were predicted as versicolor.")

pred <- prediction(set_vers_true_labels$setosa, set_vers_true_labels$Species, 
label.ordering = c("versicolor", "setosa")) 
#24 filtered PredScores for setosa column are presented in df.
#Positive/high PredScores are marked as 'setosa', otherwise, obs. with negative scores labeled as 'versicolor') (Levels: versicolor < setosa)
#Also, outputs for cutoff 'n' TP, TN, FP, FN and more.


#ROC Curve:
perf <- performance(pred,"tpr","fpr")
plot(perf, main="ROC curve for LDA prediction of setosa class:\n sensitivity vs specificty", col = "red")

Logistic Regression

\(logodds_i=B_0 + B_1X_{i1}\)

Here, the log odds represents the log odds of \(Y_i\) being 0 or 1.

Where \(logodds\) is the dependent variable, and \(X_i\) is the independent variable. \(B_{number}\) are the parameters to fit.

Logistic Regression assumes a linear relationship between the \(logodds\) and \(X\).

To convert from logodds, a not intuitive quantity, to odds, a more intuitive quantity, we use this non-linear equation:

\(odds_i=e^{logodds_{i}}\) or \(odds_i=e^{B_0 + B_1X_{i1}}\)

Odds is defined as the probability that the event will occur divided by the probability that the event will not occur.

Now we convert from odds to probability.

The probability that an event will occur is the fraction of times you expect to see that event in many trials. Probabilities always range between 0 and 1.

To convert from odds to a probability, divide the odds by one plus the odds. So to convert odds of 1/9 to a probability, divide 1/9 by 10/9 to obtain the probability of 0.10

Note: odds=a/b probty=a/(a+b)

odds <- e^(log_odds)

\(P=odds/(odds+1)\)

Logistic Regression implementation

  • Y=1 is the probability of the event occuring.
  • Independent variables should not be correlated.
  • Log odds and independent variables should be linearly correlated.
  1. Train and fit model
data(iris)

#split into training and test set 
train_size <- floor(0.75 * nrow(iris))
set.seed(543)
train_pos <- sample(seq_len(nrow(iris)), size = train_size)
train_classifier <- iris[train_pos,]
test_classifier <- iris[-train_pos,]


dim(train_classifier) #[11:5]
## [1] 112   5
dim(test_classifier) #[38:5]
## [1] 38  5
#Preparing data for Logistic Regression
#only look at two classes in each of data chunck:
train_classifier_log <- train_classifier[c(which(train_classifier$Species == "setosa"),
                                           which(train_classifier$Species == "versicolor")),]

test_classifier_log <- test_classifier[c(which(test_classifier$Species == "setosa"), 
                                         which(test_classifier$Species == "versicolor")),]

#Stashing away Predictors for each data chunck:
train_classifier_log$Species <- factor(train_classifier_log$Species)
test_classifier_log$Species <- factor(test_classifier_log$Species)

#Setting up control parameters for fitting Logistic Regression:
ctrl <- trainControl(method = "repeatedcv", repeats = 15,classProbs = T,
                     savePredictions = T)
#summary(ctrl)

#create model. logistic regression is a bionomial general linear model. 
#predict species based on sepal length with set up control
logistic_regression <- train(Species~ Sepal.Length, data = train_classifier_log, 
                             method = "glm", family= "binomial", trControl = ctrl)
logistic_regression
## Generalized Linear Model 
## 
## 76 samples
##  1 predictor
##  2 classes: 'setosa', 'versicolor' 
## 
## No pre-processing
## Resampling: Cross-Validated (10 fold, repeated 15 times) 
## Summary of sample sizes: 68, 68, 68, 69, 69, 68, ... 
## Resampling results:
## 
##   Accuracy   Kappa    
##   0.8897619  0.7782065

kappa is a coefficient of agreement, ranges [-1;+1]. 0=random chance. is rarely negative.

summary(logistic_regression)
## 
## Call:
## NULL
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.02223  -0.45039   0.03625   0.41197   2.16274  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   -28.134      6.377  -4.412 1.02e-05 ***
## Sepal.Length    5.179      1.172   4.419 9.93e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 105.15  on 75  degrees of freedom
## Residual deviance:  46.67  on 74  degrees of freedom
## AIC: 50.67
## 
## Number of Fisher Scoring iterations: 6
  1. Visualize ROC curve
plot(x = roc(predictor = logistic_regression$pred$setosa,
             response = logistic_regression$pred$obs)$specificities, 
     y = roc(predictor = logistic_regression$pred$setosa, 
             response = logistic_regression$pred$obs)$sensitivities,
     col= "blue", xlim = c(1, 0), type ="l", ylab = "Sensitivity",
     xlab = "Specificity")
## Setting levels: control = setosa, case = versicolor
## Setting direction: controls > cases
## Setting levels: control = setosa, case = versicolor
## Setting direction: controls > cases
legend("bottomright", legend = paste("setosa v versicolor --", 
                                     roc(predictor = logistic_regression$pred$setosa,
                                         response = logistic_regression$pred$obs)$auc
, sep = ""), col = c("blue"), fill = c("blue"))
## Setting levels: control = setosa, case = versicolor
## Setting direction: controls > cases

  1. Test on an independent set
#predict iris species using Sepal legth
logistic_regression_predict_class <- predict(logistic_regression, 
                                             newdata = test_classifier_log)

#confusion matrix
confusionMatrix(logistic_regression_predict_class, 
                reference = test_classifier_log$Species)
## Confusion Matrix and Statistics
## 
##             Reference
## Prediction   setosa versicolor
##   setosa         13          2
##   versicolor      1          8
##                                           
##                Accuracy : 0.875           
##                  95% CI : (0.6764, 0.9734)
##     No Information Rate : 0.5833          
##     P-Value [Acc > NIR] : 0.002161        
##                                           
##                   Kappa : 0.7391          
##                                           
##  Mcnemar's Test P-Value : 1.000000        
##                                           
##             Sensitivity : 0.9286          
##             Specificity : 0.8000          
##          Pos Pred Value : 0.8667          
##          Neg Pred Value : 0.8889          
##              Prevalence : 0.5833          
##          Detection Rate : 0.5417          
##    Detection Prevalence : 0.6250          
##       Balanced Accuracy : 0.8643          
##                                           
##        'Positive' Class : setosa          
## 

! Check if log odds and independent variables are linearly correlated

#Probability for each event for each class:
logistic_regression_predict <- predict(logistic_regression, 
                                       newdata = test_classifier_log, type = "prob")  #df[testset:2(Y1 and Y2)]

# To convert from a probability to odds, divide the probability by one minus that probability. So if the probability is 10% or 0.10 , then the odds are 0.1/0.9 or ‘1 to 9’ 
odds_species1 <- logistic_regression_predict[,1] / (1 - logistic_regression_predict[,1])
log_odds_species1 <- log(odds_species1)

#log_odds and indep.variable X must be linearly correlated:
cor.test(log_odds_species1, test_classifier_log$Sepal.Length)
## 
##  Pearson's product-moment correlation
## 
## data:  log_odds_species1 and test_classifier_log$Sepal.Length
## t = -Inf, df = 22, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -1 -1
## sample estimates:
## cor 
##  -1
plot(log_odds_species1, test_classifier_log$Sepal.Length)

Look deeper at the logistic regression

#Again, probability for each event for 2 classes:
logistic_predict_prob <- predict(logistic_regression,
                                 newdata = test_classifier_log, type="prob") #df[90:2]

#prepare df to plot = predicted + actual labels:
logistic_pred_prob_plot <- data.frame(Species_pred = logistic_predict_prob, Sepal.Length  = test_classifier_log$Sepal.Length) 

#Stash away actual labels as numeric (as logical vector):
test_classifier_log$Species <- as.numeric(test_classifier_log$Species) -1

# Plotting predictions and actual observations 
ggplot(data = test_classifier_log) +
  geom_point(aes(x=Sepal.Length, y = Species)) + 
  geom_line(data = logistic_pred_prob_plot, aes(x = Sepal.Length, 
                                                y = Species_pred.setosa, col =  "setosa"))+
  geom_line(data = logistic_pred_prob_plot, aes(x = Sepal.Length,
                                                y = Species_pred.versicolor, col = "versicolor"))+
  ggtitle("Probabilities for classifying species")

#Homework:

  1. Use the Breast Cancer dataset from the mlbench package, and predict whether the cancer is malignant or benign using one of the algorithms we learned about in class. Give some rationale as to why you chose this algorithm. Plot ROC curves, and confusion matrices. If you are choosing a hyperparameter like K or lambda, explain how and why you chose it.

Downloading data:

#install.packages("mlbench")
library(mlbench)
data(BreastCancer)
head(BreastCancer)
##        Id Cl.thickness Cell.size Cell.shape Marg.adhesion Epith.c.size
## 1 1000025            5         1          1             1            2
## 2 1002945            5         4          4             5            7
## 3 1015425            3         1          1             1            2
## 4 1016277            6         8          8             1            3
## 5 1017023            4         1          1             3            2
## 6 1017122            8        10         10             8            7
##   Bare.nuclei Bl.cromatin Normal.nucleoli Mitoses     Class
## 1           1           3               1       1    benign
## 2          10           3               2       1    benign
## 3           2           3               1       1    benign
## 4           4           3               7       1    benign
## 5           1           3               1       1    benign
## 6          10           9               7       1 malignant
str(BreastCancer)
## 'data.frame':    699 obs. of  11 variables:
##  $ Id             : chr  "1000025" "1002945" "1015425" "1016277" ...
##  $ Cl.thickness   : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 5 5 3 6 4 8 1 2 2 4 ...
##  $ Cell.size      : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 1 4 1 8 1 10 1 1 1 2 ...
##  $ Cell.shape     : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 1 4 1 8 1 10 1 2 1 1 ...
##  $ Marg.adhesion  : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 1 5 1 1 3 8 1 1 1 1 ...
##  $ Epith.c.size   : Ord.factor w/ 10 levels "1"<"2"<"3"<"4"<..: 2 7 2 3 2 7 2 2 2 2 ...
##  $ Bare.nuclei    : Factor w/ 10 levels "1","2","3","4",..: 1 10 2 4 1 10 10 1 1 1 ...
##  $ Bl.cromatin    : Factor w/ 10 levels "1","2","3","4",..: 3 3 3 3 3 9 3 3 1 2 ...
##  $ Normal.nucleoli: Factor w/ 10 levels "1","2","3","4",..: 1 2 1 7 1 7 1 1 1 1 ...
##  $ Mitoses        : Factor w/ 9 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 5 1 ...
##  $ Class          : Factor w/ 2 levels "benign","malignant": 1 1 1 1 1 2 1 1 1 1 ...
#is.na(BreastCancer) # couple of NA in $Bare.nuclei
dim(BreastCancer)
## [1] 699  11
BreastCancer_clean<- na.omit(BreastCancer)
is.na(BreastCancer_clean)
##        Id Cl.thickness Cell.size Cell.shape Marg.adhesion Epith.c.size
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## 359 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 360 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 361 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 362 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 363 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 364 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 365 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 366 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 367 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 368 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 369 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 370 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 371 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 372 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 373 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 374 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 375 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 376 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 377 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 378 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 379 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 380 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 381 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 382 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 383 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 384 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 385 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 386 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 387 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 388 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 389 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 390 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 391 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 392 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 393 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 394 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 395 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 396 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 397 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 398 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 399 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 400 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 401 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 402 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 403 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 404 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 405 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 406 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 407 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 408 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 409 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 410 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 411 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 413 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 414 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 415 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 416 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 417 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 418 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 419 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 420 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 421 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 422 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 423 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 424 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 425 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 426 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 427 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 428 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 429 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 430 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 431 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 432 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 433 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 434 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 435 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 436 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 437 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 438 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 439 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 440 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 441 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 442 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 443 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 444 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 445 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 446 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 447 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 448 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 449 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 450 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 451 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 452 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 453 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 454 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 455 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 456 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 457 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 458 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 459 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 460 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 461 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 462 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 463 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 464 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 465 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 466 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 467 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 468 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 469 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 470 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 471 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 472 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 473 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 474 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 475 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 476 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 477 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 478 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 479 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 480 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 481 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 482 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 483 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 484 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 485 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 486 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 487 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 488 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 489 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 490 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 491 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 492 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 493 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 494 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 495 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 496 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 497 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 498 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 499 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 500 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 501 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 502 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 503 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 504 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 505 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 506 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 507 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 508 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 509 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 510 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 511 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 512 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 513 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 514 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 515 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 516 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 517 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 518 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 519 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 520 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 521 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 522 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 523 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 524 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 525 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 526 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 527 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 528 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 529 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 530 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 531 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 532 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 533 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 534 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 535 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 536 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 537 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 538 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 539 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 540 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 541 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 542 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 543 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 544 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 545 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 546 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 547 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 548 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 549 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 550 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 551 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 552 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 553 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 554 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 555 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 556 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 557 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 558 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 559 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 560 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 561 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 562 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 563 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 564 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 565 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 566 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 567 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 568 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 569 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 570 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 571 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 572 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 573 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 574 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 575 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 576 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 577 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 578 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 579 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 580 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 581 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 582 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 583 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 584 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 585 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 586 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 587 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 588 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 589 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 590 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 591 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 592 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 593 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 594 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 595 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 596 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 597 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 598 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 599 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 600 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 601 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 602 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 603 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 604 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 605 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 606 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 607 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 608 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 609 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 610 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 611 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 612 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 613 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 614 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 615 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 616 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 617 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 619 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 620 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 621 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 622 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 623 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 624 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 625 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 626 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 627 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 628 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 629 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 630 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 631 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 632 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 633 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 634 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 635 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 636 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 637 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 638 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 639 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 640 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 641 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 642 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 643 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 644 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 645 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 646 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 647 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 648 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 649 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 650 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 651 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 652 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 653 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 654 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 655 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 656 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 657 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 658 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 659 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 660 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 661 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 662 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 663 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 664 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 665 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 666 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 667 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 668 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 669 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 670 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 671 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 672 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 673 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 674 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 675 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 676 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 677 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 678 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 679 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 680 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 681 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 682 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 683 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 684 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 685 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 686 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 687 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 688 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 689 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 690 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 691 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 692 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 693 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 694 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 695 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 696 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 697 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 698 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
## 699 FALSE        FALSE     FALSE      FALSE         FALSE        FALSE
##     Bare.nuclei Bl.cromatin Normal.nucleoli Mitoses Class
## 1         FALSE       FALSE           FALSE   FALSE FALSE
## 2         FALSE       FALSE           FALSE   FALSE FALSE
## 3         FALSE       FALSE           FALSE   FALSE FALSE
## 4         FALSE       FALSE           FALSE   FALSE FALSE
## 5         FALSE       FALSE           FALSE   FALSE FALSE
## 6         FALSE       FALSE           FALSE   FALSE FALSE
## 7         FALSE       FALSE           FALSE   FALSE FALSE
## 8         FALSE       FALSE           FALSE   FALSE FALSE
## 9         FALSE       FALSE           FALSE   FALSE FALSE
## 10        FALSE       FALSE           FALSE   FALSE FALSE
## 11        FALSE       FALSE           FALSE   FALSE FALSE
## 12        FALSE       FALSE           FALSE   FALSE FALSE
## 13        FALSE       FALSE           FALSE   FALSE FALSE
## 14        FALSE       FALSE           FALSE   FALSE FALSE
## 15        FALSE       FALSE           FALSE   FALSE FALSE
## 16        FALSE       FALSE           FALSE   FALSE FALSE
## 17        FALSE       FALSE           FALSE   FALSE FALSE
## 18        FALSE       FALSE           FALSE   FALSE FALSE
## 19        FALSE       FALSE           FALSE   FALSE FALSE
## 20        FALSE       FALSE           FALSE   FALSE FALSE
## 21        FALSE       FALSE           FALSE   FALSE FALSE
## 22        FALSE       FALSE           FALSE   FALSE FALSE
## 23        FALSE       FALSE           FALSE   FALSE FALSE
## 25        FALSE       FALSE           FALSE   FALSE FALSE
## 26        FALSE       FALSE           FALSE   FALSE FALSE
## 27        FALSE       FALSE           FALSE   FALSE FALSE
## 28        FALSE       FALSE           FALSE   FALSE FALSE
## 29        FALSE       FALSE           FALSE   FALSE FALSE
## 30        FALSE       FALSE           FALSE   FALSE FALSE
## 31        FALSE       FALSE           FALSE   FALSE FALSE
## 32        FALSE       FALSE           FALSE   FALSE FALSE
## 33        FALSE       FALSE           FALSE   FALSE FALSE
## 34        FALSE       FALSE           FALSE   FALSE FALSE
## 35        FALSE       FALSE           FALSE   FALSE FALSE
## 36        FALSE       FALSE           FALSE   FALSE FALSE
## 37        FALSE       FALSE           FALSE   FALSE FALSE
## 38        FALSE       FALSE           FALSE   FALSE FALSE
## 39        FALSE       FALSE           FALSE   FALSE FALSE
## 40        FALSE       FALSE           FALSE   FALSE FALSE
## 42        FALSE       FALSE           FALSE   FALSE FALSE
## 43        FALSE       FALSE           FALSE   FALSE FALSE
## 44        FALSE       FALSE           FALSE   FALSE FALSE
## 45        FALSE       FALSE           FALSE   FALSE FALSE
## 46        FALSE       FALSE           FALSE   FALSE FALSE
## 47        FALSE       FALSE           FALSE   FALSE FALSE
## 48        FALSE       FALSE           FALSE   FALSE FALSE
## 49        FALSE       FALSE           FALSE   FALSE FALSE
## 50        FALSE       FALSE           FALSE   FALSE FALSE
## 51        FALSE       FALSE           FALSE   FALSE FALSE
## 52        FALSE       FALSE           FALSE   FALSE FALSE
## 53        FALSE       FALSE           FALSE   FALSE FALSE
## 54        FALSE       FALSE           FALSE   FALSE FALSE
## 55        FALSE       FALSE           FALSE   FALSE FALSE
## 56        FALSE       FALSE           FALSE   FALSE FALSE
## 57        FALSE       FALSE           FALSE   FALSE FALSE
## 58        FALSE       FALSE           FALSE   FALSE FALSE
## 59        FALSE       FALSE           FALSE   FALSE FALSE
## 60        FALSE       FALSE           FALSE   FALSE FALSE
## 61        FALSE       FALSE           FALSE   FALSE FALSE
## 62        FALSE       FALSE           FALSE   FALSE FALSE
## 63        FALSE       FALSE           FALSE   FALSE FALSE
## 64        FALSE       FALSE           FALSE   FALSE FALSE
## 65        FALSE       FALSE           FALSE   FALSE FALSE
## 66        FALSE       FALSE           FALSE   FALSE FALSE
## 67        FALSE       FALSE           FALSE   FALSE FALSE
## 68        FALSE       FALSE           FALSE   FALSE FALSE
## 69        FALSE       FALSE           FALSE   FALSE FALSE
## 70        FALSE       FALSE           FALSE   FALSE FALSE
## 71        FALSE       FALSE           FALSE   FALSE FALSE
## 72        FALSE       FALSE           FALSE   FALSE FALSE
## 73        FALSE       FALSE           FALSE   FALSE FALSE
## 74        FALSE       FALSE           FALSE   FALSE FALSE
## 75        FALSE       FALSE           FALSE   FALSE FALSE
## 76        FALSE       FALSE           FALSE   FALSE FALSE
## 77        FALSE       FALSE           FALSE   FALSE FALSE
## 78        FALSE       FALSE           FALSE   FALSE FALSE
## 79        FALSE       FALSE           FALSE   FALSE FALSE
## 80        FALSE       FALSE           FALSE   FALSE FALSE
## 81        FALSE       FALSE           FALSE   FALSE FALSE
## 82        FALSE       FALSE           FALSE   FALSE FALSE
## 83        FALSE       FALSE           FALSE   FALSE FALSE
## 84        FALSE       FALSE           FALSE   FALSE FALSE
## 85        FALSE       FALSE           FALSE   FALSE FALSE
## 86        FALSE       FALSE           FALSE   FALSE FALSE
## 87        FALSE       FALSE           FALSE   FALSE FALSE
## 88        FALSE       FALSE           FALSE   FALSE FALSE
## 89        FALSE       FALSE           FALSE   FALSE FALSE
## 90        FALSE       FALSE           FALSE   FALSE FALSE
## 91        FALSE       FALSE           FALSE   FALSE FALSE
## 92        FALSE       FALSE           FALSE   FALSE FALSE
## 93        FALSE       FALSE           FALSE   FALSE FALSE
## 94        FALSE       FALSE           FALSE   FALSE FALSE
## 95        FALSE       FALSE           FALSE   FALSE FALSE
## 96        FALSE       FALSE           FALSE   FALSE FALSE
## 97        FALSE       FALSE           FALSE   FALSE FALSE
## 98        FALSE       FALSE           FALSE   FALSE FALSE
## 99        FALSE       FALSE           FALSE   FALSE FALSE
## 100       FALSE       FALSE           FALSE   FALSE FALSE
## 101       FALSE       FALSE           FALSE   FALSE FALSE
## 102       FALSE       FALSE           FALSE   FALSE FALSE
## 103       FALSE       FALSE           FALSE   FALSE FALSE
## 104       FALSE       FALSE           FALSE   FALSE FALSE
## 105       FALSE       FALSE           FALSE   FALSE FALSE
## 106       FALSE       FALSE           FALSE   FALSE FALSE
## 107       FALSE       FALSE           FALSE   FALSE FALSE
## 108       FALSE       FALSE           FALSE   FALSE FALSE
## 109       FALSE       FALSE           FALSE   FALSE FALSE
## 110       FALSE       FALSE           FALSE   FALSE FALSE
## 111       FALSE       FALSE           FALSE   FALSE FALSE
## 112       FALSE       FALSE           FALSE   FALSE FALSE
## 113       FALSE       FALSE           FALSE   FALSE FALSE
## 114       FALSE       FALSE           FALSE   FALSE FALSE
## 115       FALSE       FALSE           FALSE   FALSE FALSE
## 116       FALSE       FALSE           FALSE   FALSE FALSE
## 117       FALSE       FALSE           FALSE   FALSE FALSE
## 118       FALSE       FALSE           FALSE   FALSE FALSE
## 119       FALSE       FALSE           FALSE   FALSE FALSE
## 120       FALSE       FALSE           FALSE   FALSE FALSE
## 121       FALSE       FALSE           FALSE   FALSE FALSE
## 122       FALSE       FALSE           FALSE   FALSE FALSE
## 123       FALSE       FALSE           FALSE   FALSE FALSE
## 124       FALSE       FALSE           FALSE   FALSE FALSE
## 125       FALSE       FALSE           FALSE   FALSE FALSE
## 126       FALSE       FALSE           FALSE   FALSE FALSE
## 127       FALSE       FALSE           FALSE   FALSE FALSE
## 128       FALSE       FALSE           FALSE   FALSE FALSE
## 129       FALSE       FALSE           FALSE   FALSE FALSE
## 130       FALSE       FALSE           FALSE   FALSE FALSE
## 131       FALSE       FALSE           FALSE   FALSE FALSE
## 132       FALSE       FALSE           FALSE   FALSE FALSE
## 133       FALSE       FALSE           FALSE   FALSE FALSE
## 134       FALSE       FALSE           FALSE   FALSE FALSE
## 135       FALSE       FALSE           FALSE   FALSE FALSE
## 136       FALSE       FALSE           FALSE   FALSE FALSE
## 137       FALSE       FALSE           FALSE   FALSE FALSE
## 138       FALSE       FALSE           FALSE   FALSE FALSE
## 139       FALSE       FALSE           FALSE   FALSE FALSE
## 141       FALSE       FALSE           FALSE   FALSE FALSE
## 142       FALSE       FALSE           FALSE   FALSE FALSE
## 143       FALSE       FALSE           FALSE   FALSE FALSE
## 144       FALSE       FALSE           FALSE   FALSE FALSE
## 145       FALSE       FALSE           FALSE   FALSE FALSE
## 147       FALSE       FALSE           FALSE   FALSE FALSE
## 148       FALSE       FALSE           FALSE   FALSE FALSE
## 149       FALSE       FALSE           FALSE   FALSE FALSE
## 150       FALSE       FALSE           FALSE   FALSE FALSE
## 151       FALSE       FALSE           FALSE   FALSE FALSE
## 152       FALSE       FALSE           FALSE   FALSE FALSE
## 153       FALSE       FALSE           FALSE   FALSE FALSE
## 154       FALSE       FALSE           FALSE   FALSE FALSE
## 155       FALSE       FALSE           FALSE   FALSE FALSE
## 156       FALSE       FALSE           FALSE   FALSE FALSE
## 157       FALSE       FALSE           FALSE   FALSE FALSE
## 158       FALSE       FALSE           FALSE   FALSE FALSE
## 160       FALSE       FALSE           FALSE   FALSE FALSE
## 161       FALSE       FALSE           FALSE   FALSE FALSE
## 162       FALSE       FALSE           FALSE   FALSE FALSE
## 163       FALSE       FALSE           FALSE   FALSE FALSE
## 164       FALSE       FALSE           FALSE   FALSE FALSE
## 166       FALSE       FALSE           FALSE   FALSE FALSE
## 167       FALSE       FALSE           FALSE   FALSE FALSE
## 168       FALSE       FALSE           FALSE   FALSE FALSE
## 169       FALSE       FALSE           FALSE   FALSE FALSE
## 170       FALSE       FALSE           FALSE   FALSE FALSE
## 171       FALSE       FALSE           FALSE   FALSE FALSE
## 172       FALSE       FALSE           FALSE   FALSE FALSE
## 173       FALSE       FALSE           FALSE   FALSE FALSE
## 174       FALSE       FALSE           FALSE   FALSE FALSE
## 175       FALSE       FALSE           FALSE   FALSE FALSE
## 176       FALSE       FALSE           FALSE   FALSE FALSE
## 177       FALSE       FALSE           FALSE   FALSE FALSE
## 178       FALSE       FALSE           FALSE   FALSE FALSE
## 179       FALSE       FALSE           FALSE   FALSE FALSE
## 180       FALSE       FALSE           FALSE   FALSE FALSE
## 181       FALSE       FALSE           FALSE   FALSE FALSE
## 182       FALSE       FALSE           FALSE   FALSE FALSE
## 183       FALSE       FALSE           FALSE   FALSE FALSE
## 184       FALSE       FALSE           FALSE   FALSE FALSE
## 185       FALSE       FALSE           FALSE   FALSE FALSE
## 186       FALSE       FALSE           FALSE   FALSE FALSE
## 187       FALSE       FALSE           FALSE   FALSE FALSE
## 188       FALSE       FALSE           FALSE   FALSE FALSE
## 189       FALSE       FALSE           FALSE   FALSE FALSE
## 190       FALSE       FALSE           FALSE   FALSE FALSE
## 191       FALSE       FALSE           FALSE   FALSE FALSE
## 192       FALSE       FALSE           FALSE   FALSE FALSE
## 193       FALSE       FALSE           FALSE   FALSE FALSE
## 194       FALSE       FALSE           FALSE   FALSE FALSE
## 195       FALSE       FALSE           FALSE   FALSE FALSE
## 196       FALSE       FALSE           FALSE   FALSE FALSE
## 197       FALSE       FALSE           FALSE   FALSE FALSE
## 198       FALSE       FALSE           FALSE   FALSE FALSE
## 199       FALSE       FALSE           FALSE   FALSE FALSE
## 200       FALSE       FALSE           FALSE   FALSE FALSE
## 201       FALSE       FALSE           FALSE   FALSE FALSE
## 202       FALSE       FALSE           FALSE   FALSE FALSE
## 203       FALSE       FALSE           FALSE   FALSE FALSE
## 204       FALSE       FALSE           FALSE   FALSE FALSE
## 205       FALSE       FALSE           FALSE   FALSE FALSE
## 206       FALSE       FALSE           FALSE   FALSE FALSE
## 207       FALSE       FALSE           FALSE   FALSE FALSE
## 208       FALSE       FALSE           FALSE   FALSE FALSE
## 209       FALSE       FALSE           FALSE   FALSE FALSE
## 210       FALSE       FALSE           FALSE   FALSE FALSE
## 211       FALSE       FALSE           FALSE   FALSE FALSE
## 212       FALSE       FALSE           FALSE   FALSE FALSE
## 213       FALSE       FALSE           FALSE   FALSE FALSE
## 214       FALSE       FALSE           FALSE   FALSE FALSE
## 215       FALSE       FALSE           FALSE   FALSE FALSE
## 216       FALSE       FALSE           FALSE   FALSE FALSE
## 217       FALSE       FALSE           FALSE   FALSE FALSE
## 218       FALSE       FALSE           FALSE   FALSE FALSE
## 219       FALSE       FALSE           FALSE   FALSE FALSE
## 220       FALSE       FALSE           FALSE   FALSE FALSE
## 221       FALSE       FALSE           FALSE   FALSE FALSE
## 222       FALSE       FALSE           FALSE   FALSE FALSE
## 223       FALSE       FALSE           FALSE   FALSE FALSE
## 224       FALSE       FALSE           FALSE   FALSE FALSE
## 225       FALSE       FALSE           FALSE   FALSE FALSE
## 226       FALSE       FALSE           FALSE   FALSE FALSE
## 227       FALSE       FALSE           FALSE   FALSE FALSE
## 228       FALSE       FALSE           FALSE   FALSE FALSE
## 229       FALSE       FALSE           FALSE   FALSE FALSE
## 230       FALSE       FALSE           FALSE   FALSE FALSE
## 231       FALSE       FALSE           FALSE   FALSE FALSE
## 232       FALSE       FALSE           FALSE   FALSE FALSE
## 233       FALSE       FALSE           FALSE   FALSE FALSE
## 234       FALSE       FALSE           FALSE   FALSE FALSE
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dim(BreastCancer_clean) [683:11]
##   [1] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
##  [24] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
##  [47] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
##  [70] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
##  [93] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [116] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [139] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [162] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [185] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [208] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [231] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [254] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [277] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [300] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [323] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [346] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [369] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [392] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [415] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [438] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [461] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [484] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [507] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [530] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [553] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [576] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [599] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [622] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [645] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## [668] NA NA NA NA NA NA
summary(BreastCancer_clean$Class)
##    benign malignant 
##       444       239

Data:

n=699, after NA ommited, n=683 9 variables - ordinary numbers with 10 or 9 levels each 458-B and 241-M. After cleaning, predictor’s 2 classes:444-B and 239-M

The main goal of this task is to find a function to classify accurately predictor’s 2 classes.

Method chosen is Dimension Reduction with Multiple Correspondence Analysis and Logistic Regression build on selected dimension/features:

  1. Since the variables are all categorical, I will use MCA as a dimension reduction technique. MCA enables the study of both the relationship between the observations and the associations between variable categories (biplot).
    {The other technique for me to try is ‘latent class analysis in R’ next time.}

{If I had more than two classes, I would have considered LDA as linear classification and DR method for categorical data.}

  1. MCA raw coordinates from selected dimension(s) will serve as a new predictor for Logistic Regression on train set to build the classification model. Afterwards, the model will be validated on test set.

3.Depending on the model performance (ROC and AUC), I might proceed for L1 and L2 regularizations.

1. Multiple Correspondence Analysis (MCA) for categ.variables without predictor:

(http://www.sthda.com/english/wiki/print.php?id=232#variable-categories)

data(BreastCancer_clean)
## Warning in data(BreastCancer_clean): data set 'BreastCancer_clean' not
## found
head( BreastCancer_MCA<- BreastCancer_clean[, 2:10])  #excl.Ind and Class
##   Cl.thickness Cell.size Cell.shape Marg.adhesion Epith.c.size Bare.nuclei
## 1            5         1          1             1            2           1
## 2            5         4          4             5            7          10
## 3            3         1          1             1            2           2
## 4            6         8          8             1            3           4
## 5            4         1          1             3            2           1
## 6            8        10         10             8            7          10
##   Bl.cromatin Normal.nucleoli Mitoses
## 1           3               1       1
## 2           3               2       1
## 3           3               1       1
## 4           3               7       1
## 5           3               1       1
## 6           9               7       1

Data: Plotting the frequency of variable categories:

for (i in 1:ncol(BreastCancer_MCA)) {
  plot(BreastCancer_MCA[,i], main=colnames(BreastCancer_MCA)[i],
       ylab = "Count", col="steelblue", las = 2)
  }

*Optional: Histogram of variable counts (though I am not using LDA here)

cellsize <- ggplot(data = BreastCancer_clean, aes(x = as.numeric(Cell.size), fill = Class)) + 
  geom_histogram(position="identity", alpha=0.5, bins= 25) +
  ggtitle("Cell size by Class (B/M)")+
  theme_bw()
cellsize

cellshape <- ggplot(data = BreastCancer_clean, aes(x = as.numeric(Cell.shape), fill = Class)) + 
  geom_histogram(position="identity", alpha=0.5, bins= 25) +
  ggtitle("Cell shape by Class (B/M)")+
  theme_bw()
cellshape

grid.arrange(cellsize, cellshape)

####MCA: FactoMineR::MCA() (by default: MCA(X, ncp = 5, graph = TRUE)) X : a data frame with n rows (individuals) and p columns (categorical variables) ncp : number of dimensions kept in the final results. graph : a logical value. If TRUE a graph is displayed.

(guidance from: https://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-td3750251.html)

#Then, I used mjca of ca pacakge for MCA. 
#install.packages("ca")
library(ca)
mjca1 <-  mjca(BreastCancer_MCA) 

summary(mjca1) 
## 
## Principal inertias (eigenvalues):
## 
##  dim    value      %   cum%   scree plot               
##  1      0.470641  65.8  65.8  *******************      
##  2      0.050913   7.1  72.9  **                       
##  3      0.018727   2.6  75.5  *                        
##  4      0.012462   1.7  77.2                           
##  5      0.011066   1.5  78.8                           
##  6      0.009880   1.4  80.2                           
##  7      0.008198   1.1  81.3                           
##  8      0.007278   1.0  82.3                           
##  9      0.006711   0.9  83.3                           
##  10     0.005891   0.8  84.1                           
##  11     0.004856   0.7  84.8                           
##  12     0.004246   0.6  85.4                           
##  13     0.003943   0.6  85.9                           
##  14     0.003128   0.4  86.3                           
##  15     0.002650   0.4  86.7                           
##  16     0.002494   0.3  87.1                           
##  17     0.001875   0.3  87.3                           
##  18     0.001701   0.2  87.6                           
##  19     0.001434   0.2  87.8                           
##  20     0.001263   0.2  87.9                           
##  21     0.001111   0.2  88.1                           
##  22     0.000991   0.1  88.2                           
##  23     0.000653   0.1  88.3                           
##  24     0.000525   0.1  88.4                           
##  25     0.000448   0.1  88.5                           
##  26     0.000254   0.0  88.5                           
##  27     0.000186   0.0  88.5                           
##  28     0.000154   0.0  88.5                           
##  29     9.4e-050   0.0  88.6                           
##  30     5.7e-050   0.0  88.6                           
##  31     5.5e-050   0.0  88.6                           
##  32     3e-05000   0.0  88.6                           
##  33     1.4e-050   0.0  88.6                           
##         -------- -----                                 
##  Total: 0.715685                                       
## 
## 
## Columns:
##                    name   mass  qlt  inr     k=1 cor ctr    k=2 cor ctr  
## 1  |     Cl.thickness:1 |   23  952   11 |   617 937  18 |   79  15   3 |
## 2  |     Cl.thickness:2 |    8  812    9 |   524 811   5 |   21   1   0 |
## 3  |     Cl.thickness:3 |   17  816    8 |   400 805   6 |  -48  11   1 |
## 4  |     Cl.thickness:4 |   13  689    8 |   360 686   4 |  -24   3   0 |
## 5  |     Cl.thickness:5 |   21   30    7 |   -43  30   0 |    0   0   0 |
## 6  |     Cl.thickness:6 |    5  489    9 |  -561 478   4 |   83  11   1 |
## 7  |     Cl.thickness:7 |    4  699   11 | -1032 529   8 | -583 169  25 |
## 8  |     Cl.thickness:8 |    7  786   13 | -1042 785  17 |  -22   0   0 |
## 9  |     Cl.thickness:9 |    2  517   11 | -1088 472   6 |  335  45   5 |
## 10 |    Cl.thickness:10 |   11  903   14 | -1060 903  27 |   26   1   0 |
## 11 |        Cell.size:1 |   61  952   17 |   612 942  48 |   64  10   5 |
## 12 |        Cell.size:2 |    7  270    9 |   297 202   1 | -172  68   4 |
## 13 |        Cell.size:3 |    8  404    9 |  -218  93   1 | -400 311  27 |
## 14 |        Cell.size:4 |    6  588   12 |  -764 418   8 | -487 170  29 |
## 15 |        Cell.size:5 |    5  624   13 | -1076 550  12 | -394  74  15 |
## 16 |        Cell.size:6 |    4  561   12 | -1011 502   9 | -346  59  10 |
## 17 |        Cell.size:7 |    3  538   12 | -1137 523   8 | -194  15   2 |
## 18 |        Cell.size:8 |    5  562   13 | -1109 550  12 | -167  13   3 |
## 19 |        Cell.size:9 |    1  324   10 | -1240 321   3 | -116   3   0 |
## 20 |       Cell.size:10 |   11  892   21 | -1247 638  36 |  787 254 133 |
## 21 |       Cell.shape:1 |   56  953   16 |   622 942  46 |   68  11   5 |
## 22 |       Cell.shape:2 |    9  274    9 |   294 231   2 | -128  44   3 |
## 23 |       Cell.shape:3 |    9  230    8 |   -83  25   0 | -240 205  10 |
## 24 |       Cell.shape:4 |    7  526   11 |  -595 389   5 | -354 137  17 |
## 25 |       Cell.shape:5 |    5  640   12 |  -896 523   9 | -423 117  18 |
## 26 |       Cell.shape:6 |    5  634   12 |  -925 507   9 | -464 128  20 |
## 27 |       Cell.shape:7 |    5  657   13 | -1148 586  14 | -397  70  15 |
## 28 |       Cell.shape:8 |    4  577   12 | -1106 575  11 |   70   2   0 |
## 29 |       Cell.shape:9 |    1  321   10 | -1078 321   3 |   31   0   0 |
## 30 |      Cell.shape:10 |    9  894   21 | -1289 627  33 |  841 267 131 |
## 31 |    Marg.adhesion:1 |   64  991   13 |   522 986  37 |   36   5   2 |
## 32 |    Marg.adhesion:2 |    9  183    8 |   -71  28   0 | -166 154   5 |
## 33 |    Marg.adhesion:3 |    9  363    8 |  -257 285   1 | -134  77   3 |
## 34 |    Marg.adhesion:4 |    5  735   11 |  -788 515   7 | -515 220  28 |
## 35 |    Marg.adhesion:5 |    4  661   11 | -1057 596   9 | -351  66   9 |
## 36 |    Marg.adhesion:6 |    3  605   10 |  -874 555   6 | -261  50   5 |
## 37 |    Marg.adhesion:7 |    2  457   12 | -1327 440   8 |  257  17   3 |
## 38 |    Marg.adhesion:8 |    4  689   12 | -1193 661  12 |  242  27   5 |
## 39 |    Marg.adhesion:9 |    1  198    9 |  -889 198   1 |  -21   0   0 |
## 40 |   Marg.adhesion:10 |    9  846   17 | -1212 746  28 |  443 100  35 |
## 41 |     Epith.c.size:1 |    7  846    9 |   641 826   6 |  100  20   1 |
## 42 |     Epith.c.size:2 |   61  980   13 |   540 979  38 |   17   1   0 |
## 43 |     Epith.c.size:3 |   12  661   10 |  -436 361   5 | -398 300  36 |
## 44 |     Epith.c.size:4 |    8  792   12 |  -912 692  14 | -346  99  18 |
## 45 |     Epith.c.size:5 |    6  692   12 |  -978 672  13 | -168  20   4 |
## 46 |     Epith.c.size:6 |    7  806   13 | -1082 774  16 |  222  33   6 |
## 47 |     Epith.c.size:7 |    2  483   11 | -1220 462   6 |  260  21   2 |
## 48 |     Epith.c.size:8 |    3  638   11 | -1096 589   9 |  318  50   7 |
## 49 |     Epith.c.size:9 |    0  180   10 | -1451 141   1 |  762  39   4 |
## 50 |    Epith.c.size:10 |    5  817   13 | -1149 611  14 |  667 206  44 |
## 51 |      Bare.nuclei:1 |   65  982   14 |   550 973  42 |   52   9   3 |
## 52 |      Bare.nuclei:2 |    5    9    8 |   -49   8   0 |  -17   1   0 |
## 53 |      Bare.nuclei:3 |    5  241    9 |  -242 118   1 | -247 123   5 |
## 54 |      Bare.nuclei:4 |    3  318    9 |  -557 318   2 |  -14   0   0 |
## 55 |      Bare.nuclei:5 |    5  516    9 |  -610 514   4 |  -30   1   0 |
## 56 |      Bare.nuclei:6 |    1  104   10 |  -761 103   1 |   70   1   0 |
## 57 |      Bare.nuclei:7 |    1  371   10 |  -938 232   2 | -725 139  13 |
## 58 |      Bare.nuclei:8 |    3  522   11 |  -996 509   7 | -159  13   2 |
## 59 |      Bare.nuclei:9 |    1  380   10 | -1116 376   4 | -107   3   0 |
## 60 |     Bare.nuclei:10 |   21  934   21 | -1081 934  53 |  -19   0   0 |
## 61 |      Bl.cromatin:1 |   24  946   11 |   587 940  18 |   44   5   1 |
## 62 |      Bl.cromatin:2 |   26  950   11 |   568 946  18 |   39   4   1 |
## 63 |      Bl.cromatin:3 |   26  663    7 |   232 615   3 |  -65  48   2 |
## 64 |      Bl.cromatin:4 |    6  642   11 |  -864 605  10 | -215  38   6 |
## 65 |      Bl.cromatin:5 |    6  555   11 |  -804 521   8 | -204  34   5 |
## 66 |      Bl.cromatin:6 |    1  555    9 |  -890 386   2 | -588 169  10 |
## 67 |      Bl.cromatin:7 |   12  851   15 | -1048 827  27 | -179  24   7 |
## 68 |      Bl.cromatin:8 |    5  706   13 | -1226 648  15 |  366  58  12 |
## 69 |      Bl.cromatin:9 |    2  463   11 | -1206 453   6 |  172   9   1 |
## 70 |     Bl.cromatin:10 |    3  811   13 | -1268 523  11 |  940 288  56 |
## 71 |  Normal.nucleoli:1 |   70  984   12 |   485 981  35 |   29   4   1 |
## 72 |  Normal.nucleoli:2 |    6  107    8 |   120  51   0 | -126  56   2 |
## 73 |  Normal.nucleoli:3 |    7  507   10 |  -644 462   6 | -201  45   5 |
## 74 |  Normal.nucleoli:4 |    3  535   10 |  -820 385   4 | -512 150  15 |
## 75 |  Normal.nucleoli:5 |    3  610   10 |  -977 575   6 | -241  35   4 |
## 76 |  Normal.nucleoli:6 |    4  559   10 |  -754 456   4 | -357 103   9 |
## 77 |  Normal.nucleoli:7 |    3  595   11 | -1155 593   7 |  -64   2   0 |
## 78 |  Normal.nucleoli:8 |    4  692   11 | -1129 673  10 | -190  19   3 |
## 79 |  Normal.nucleoli:9 |    2  517   11 | -1085 489   6 | -258  28   3 |
## 80 | Normal.nucleoli:10 |   10  928   18 | -1267 793  33 |  522 135  52 |
## 81 |          Mitoses:1 |   92  982    4 |   227 979  10 |  -12   3   0 |
## 82 |          Mitoses:2 |    6  741   10 |  -796 609   8 | -372 133  15 |
## 83 |          Mitoses:3 |    5  772   12 | -1144 768  15 |  -74   3   1 |
## 84 |          Mitoses:4 |    2  525   11 | -1268 512   7 | -206  13   2 |
## 85 |          Mitoses:5 |    1  339    9 |  -846 239   1 |  547 100   6 |
## 86 |          Mitoses:6 |    0  292   10 | -1446 274   2 |  371  18   1 |
## 87 |          Mitoses:7 |    1  528   10 | -1187 452   4 |  488  76   7 |
## 88 |          Mitoses:8 |    1  306   10 | -1030 301   3 |  140   6   0 |
## 89 |         Mitoses:10 |    2  751   12 | -1341 463   9 | 1058 288  50 |
# Dim1 explains 65.8% of variance, of the highest dim score on scree plot. 33 dimensions explain ~88.6% of variance.


# which values I could use like PC1 in PCA?
#"rowcoord" in mjca1 will help:. 
plot.mjca(mjca1)

mjca1$rowpcoord 
##               [,1]         [,2]          [,3]          [,4]          [,5]
##   [1,]  0.60670236  0.093148317 -0.0194645763 -0.0339920478 -0.0679413532
##   [2,] -0.67713460 -0.568312780 -0.5710272022 -0.9223240981  0.7957212383
##   [3,]  0.58140352  0.035735465 -0.0517082388 -0.0219969529  0.1303154817
##   [4,] -0.63869859 -0.261121408  1.0222161072  0.8210721236  1.3059904813
##   [5,]  0.54587161 -0.002697945  0.0940041356  0.0067468532  0.0260952186
##   [6,] -1.49085120  1.076066095  0.5478778388 -1.0914567387  0.7082658028
##   [7,]  0.44934524  0.096811484  0.0192649136 -0.1025567916 -0.1379838327
##   [8,]  0.64547145  0.007046507 -0.0647849879  0.0095033584  0.0166632459
##   [9,]  0.58228667  0.432198770 -0.0266168335  0.2753872274 -0.3994286278
##  [10,]  0.67556178  0.016013325  0.0002533900 -0.0189274323  0.2017063942
##  [11,]  0.72999692  0.172657736  0.1763149161 -0.0476324042 -0.1648406639
##  [12,]  0.75301983  0.154444713  0.1261477196 -0.0122699770 -0.1436747828
##  [13,] -0.28505505 -0.857740287 -0.5309088085 -0.4663524368  0.0648873933
##  [14,]  0.58527182 -0.015448150 -0.0091486054 -0.1495432357  0.0533481457
##  [15,] -1.56627214 -0.341420199  0.0647221833 -2.4527460932  0.3527272050
##  [16,] -0.86202769 -1.085312369 -0.8735996198  0.2212106150 -0.0626975817
##  [17,]  0.72650889  0.132198496  0.1323574802 -0.0148536952  0.0136752641
##  [18,]  0.67199796  0.081089342  0.0932921609 -0.0212693150  0.0135736037
##  [19,] -1.19663578 -0.862123854  0.7394190742 -0.2515589968 -0.3817647153
##  [20,]  0.52280977  0.133962448  0.1413363176  0.1033050009 -0.1662691630
##  [21,] -1.15281251 -0.874777363 -0.2946183542 -1.8235180282 -0.1501872060
##  [22,] -1.19800413 -0.540289385 -0.1907416530  1.5137672750 -1.1333385536
##  [23,]  0.73298481  0.120624400  0.0993995169 -0.0312889779 -0.0656680058
##  [24,]  0.71355444  0.131679912  0.0979865935 -0.0348557455 -0.0934914405
##  [25,] -0.21214729 -1.018734728 -0.7905925237  1.0354170147 -0.1846379621
##  [26,]  0.69848019  0.045417052  0.0456237493 -0.0481393738  0.0510139009
##  [27,]  0.66121329  0.144257471  0.0196007430 -0.0275764280 -0.0678396928
##  [28,]  0.75301983  0.154444713  0.1261477196 -0.0122699770 -0.1436747828
##  [29,]  0.65699386  0.034216507 -0.0384227072  0.0575957224 -0.0775981423
##  [30,]  0.74942729  0.161602224  0.1777278395 -0.0440656366 -0.1370172292
##  [31,]  0.69850890  0.103335559  0.0870824003 -0.0186855968 -0.1437764432
##  [32,] -1.34753758 -0.579352655  0.8645505009 -0.6451538713 -0.4917263196
##  [33,]  0.60246771  0.003614281 -0.0268722617  0.0485829372 -0.0775245307
##  [34,]  0.67994736  0.024335348 -0.0524678713 -0.0031000227  0.0947716833
##  [35,]  0.75301983  0.154444713  0.1261477196 -0.0122699770 -0.1436747828
##  [36,] -1.19928144  1.115978958  0.6032818339  0.6364387608 -0.5754629039
##  [37,]  0.28103339  0.002771525  0.2726454433  0.2024663528 -0.2180207989
##  [38,] -0.67409211 -0.707209574 -1.4865931427 -0.5081146686  0.4388594933
##  [39,] -0.76269814 -0.827584223 -0.2580280635  1.4050156830 -0.9225944680
##  [40,] -0.76557290 -1.235835840 -1.2510904382  0.5102855643  0.0440764264
##  [41,] -1.32504105  0.685068295 -0.7094217557  0.2082749015 -0.5824656176
##  [42,] -0.40149342 -0.176970919 -0.3179798727 -0.0705243658 -0.5703815060
##  [43,] -1.16547582  1.174128949 -0.0923094513  0.4893341863 -0.3444874362
##  [44,]  0.60235398  0.005591087  0.1348065281  0.0695779322 -0.0142783810
##  [45,] -1.04752498 -0.997038475  1.4455975265 -0.6010148998 -0.0554086676
##  [46,]  0.76806537  0.182789067  0.1370519128 -0.0284401257 -0.0933897801
##  [47,]  0.54587161 -0.002697945  0.0940041356  0.0067468532  0.0260952186
##  [48,] -1.12752916 -1.204350531  1.5537117701  0.4495315019 -0.5236469425
##  [49,] -0.36319281  0.212563584  0.1729042305  0.3448770976 -0.1538549215
##  [50,] -0.24471053 -0.856786151 -0.3034999739  0.1074755982  0.2764548558
##  [51,] -1.01206567 -0.737747325 -0.7981037304  0.3565007136  0.4728350881
##  [52,] -1.33324364  0.020728134  0.1762860867  0.4063980836 -2.2298344975
##  [53,] -1.02599515 -0.519759506  0.2506751218  0.6180498527 -2.0732728476
##  [54,] -0.82123463 -0.851552591 -0.3784235893 -0.0468259591 -0.0551705706
##  [55,] -0.79029253  0.482674176  0.1109482852 -0.1376791174  0.6619527097
##  [56,] -0.67004320 -0.762801798  0.1385277442 -1.9745951640  0.4240294871
##  [57,] -0.25296295 -0.176351981 -0.5203539717 -0.2187545130  0.2572517065
##  [58,] -0.44256403 -0.411167295 -0.5021657637 -0.0610794102 -0.4662254691
##  [59,] -0.77680060 -0.750391702 -0.6605519111 -0.1775091453  0.3068448174
##  [60,]  0.67099501  0.149009286  0.0250094765 -0.0127324809  0.1026953678
##  [61,] -0.87983618  1.098364586 -0.2751003774  0.1866747350 -0.4040600962
##  [62,] -0.40581164 -0.568284850  0.0808015429 -0.2743998273  0.2207498092
##  [63,]  0.76806537  0.182789067  0.1370519128 -0.0284401257 -0.0933897801
##  [64,] -0.70355971 -0.160073616 -1.4170131630  0.6655927114  0.5468603573
##  [65,]  0.67199796  0.081089342  0.0932921609 -0.0212693150  0.0135736037
##  [66,] -0.68566395 -0.444697860 -0.4810078119 -0.3316187644 -0.0874634322
##  [67,] -1.29453228 -0.340163492  1.4346941192  0.9383469477  1.1221912107
##  [68,]  0.65450148  0.055225029  0.0410473068 -0.0127738896  0.1611627721
##  [69,]  0.54708006 -0.006963189 -0.1749883004  0.0498606006 -0.0268181069
##  [70,] -1.20212192  1.144613980  0.2211526053  0.5420069758 -0.2860398916
##  [71,]  0.10265166 -0.480123306 -0.3286226169  0.2617462748  0.2092719569
##  [72,] -1.27778284 -0.170162675 -0.5955382823 -0.2766519897 -0.5822593158
##  [73,] -0.63658860 -0.647363024 -0.5944146654  0.1808972368  0.9917033184
##  [74,]  0.32683678 -0.148960588 -0.3390629661 -0.0608180148  0.5100056142
##  [75,]  0.57102394 -0.024695171 -0.0767455712 -0.2133031026  0.1715806759
##  [76,]  0.43076845 -0.183886628 -0.3405650306  0.0534949600  0.1076814488
##  [77,]  0.55019126 -0.077612817 -0.0468010014 -0.1523920879  0.0810699200
##  [78,]  0.59492291 -0.049647252  0.0015364367  0.1525654832 -0.0476284988
##  [79,]  0.40369507 -0.124144416  0.0665241378  0.1086647104 -0.0350883900
##  [80,]  0.63046769  0.032477219  0.0184028182  0.0524148387  0.0799271765
##  [81,]  0.55575525 -0.023036855 -0.1515686666 -0.0380657849  0.1200897770
##  [82,]  0.37413177 -0.020248110  0.1333766546  0.0940147949 -0.0381180610
##  [83,] -1.41383767  0.219464348  1.2055538674  0.0316442639 -1.7813081135
##  [84,] -1.11296882  1.170525472 -0.7409754850  0.6011408328  0.5517056017
##  [85,] -0.80375443 -1.227764338  0.2738090008 -0.5691263388 -0.6676134828
##  [86,] -1.23621740 -1.063203284 -0.0293001765 -0.5744710549 -0.8150035784
##  [87,]  0.67199796  0.081089342  0.0932921609 -0.0212693150  0.0135736037
##  [88,]  0.49888171 -0.149368529 -0.1124182253  0.2198340171  0.0186234137
##  [89,]  0.71355444  0.131679912  0.0979865935 -0.0348557455 -0.0934914405
##  [90,]  0.64000533  0.023551223  0.0045592783  0.0445783756 -0.0246460415
##  [91,]  0.67199796  0.081089342  0.0932921609 -0.0212693150  0.0135736037
##  [92,]  0.76806537  0.182789067  0.1370519128 -0.0284401257 -0.0933897801
##  [93,]  0.69850890  0.103335559  0.0870824003 -0.0186855968 -0.1437764432
##  [94,]  0.71355444  0.131679912  0.0979865935 -0.0348557455 -0.0934914405
##  [95,]  0.66004036  0.057371537  0.0313074811  0.0635973765 -0.1026528185
##  [96,]  0.60670236  0.093148317 -0.0194645763 -0.0339920478 -0.0679413532
##  [97,] -1.13622829  0.704319867  0.4890650625 -0.1681369702  1.2909441293
##  [98,] -1.57175729 -0.900417428  0.3988480758 -0.7825604101 -1.1556725765
##  [99,] -0.84919222 -0.019423162 -0.5906993655  0.2985847808  0.0667560797
## [100,] -0.21528613 -0.726340778 -0.5150282519 -0.1880270219  0.3742674736
## [101,]  0.61896050 -0.015199710 -0.0585752274  0.0069196402  0.1740132928
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## [197,]  0.71355444  0.131679912  0.0979865935 -0.0348557455 -0.0934914405
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## [290,]  0.71355444  0.131679912  0.0979865935 -0.0348557455 -0.0934914405
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## [545,]  0.66121329  0.144257471  0.0196007430 -0.0275764280 -0.0678396928
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## [549,]  0.73298481  0.120624400  0.0993995169 -0.0312889779 -0.0656680058
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## [552,]  0.62543643 -0.026773807 -0.0915331907 -0.0095156425  0.0946700229
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## [555,] -1.29601560  1.105467244 -0.4734902809  0.0379092145  0.4213638433
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## [558,]  0.73298481  0.120624400  0.0993995169 -0.0312889779 -0.0656680058
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## [560,] -1.07585170 -0.609972119  1.2780714420  0.4298004140  0.6080073138
## [561,]  0.55366490 -0.003140735 -0.1713319646 -0.0058030926  0.0924983359
## [562,]  0.66121329  0.144257471  0.0196007430 -0.0275764280 -0.0678396928
## [563,]  0.76806537  0.182789067  0.1370519128 -0.0284401257 -0.0933897801
## [564,]  0.76806537  0.182789067  0.1370519128 -0.0284401257 -0.0933897801
## [565,]  0.71355444  0.131679912  0.0979865935 -0.0348557455 -0.0934914405
## [566,]  0.60817583  0.047968420 -0.1322666452  0.0006125272  0.0925999963
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## [573,]  0.60216033  0.067802588 -0.0373385437 -0.0054945722  0.1868145198
## [574,] -0.91784546 -0.002726444  0.2653396544 -0.0369260910  1.1450676945
## [575,]  0.66427501  0.146905572  0.0387151664 -0.0189776085 -0.0930696410
## [576,] -0.64250575  0.563576589  0.4622704353 -0.2188939027  0.6400683453
## [577,] -0.99475796 -0.957769313  0.5414847467  0.4600071509 -1.5716185706
## [578,] -0.74504562 -0.834308506 -0.6262760768 -0.1795161806  0.5640884635
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## [581,]  0.66121329  0.144257471  0.0196007430 -0.0275764280 -0.0678396928
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## [586,]  0.73298481  0.120624400  0.0993995169 -0.0312889779 -0.0656680058
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## [588,]  0.72650889  0.132198496  0.1323574802 -0.0148536952  0.0136752641
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## [590,] -0.77031944 -0.221366407  0.2986696245  0.0319045272 -0.2038150742
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## [602,]  0.73298481  0.120624400  0.0993995169 -0.0312889779 -0.0656680058
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## [670,]  0.77112709  0.185437167  0.1561663363 -0.0198413062 -0.1186197283
## [671,]  0.77112709  0.185437167  0.1561663363 -0.0198413062 -0.1186197283
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## [677,]  0.73604653  0.123272501  0.1185139403 -0.0226901584 -0.0908979540
## [678,]  0.56727342 -0.056573580  0.0971541321  0.0667290800  0.0134433933
## [679,]  0.48087925 -0.114599245 -0.1181397790  0.1578529465  0.2012334779
## [680,]  0.75608155  0.157092814  0.1452621431 -0.0036711575 -0.1689047310
## [681,] -1.22877226  0.996442365 -0.1443238579 -0.2548140578  0.3154939290
## [682,] -0.85025572 -0.498306146 -0.1994170231  0.6354649292  0.4301394311
## [683,] -1.01936871 -0.213082090  1.1645502330 -0.5083030515  0.8548911073
##                 [,6]          [,7]          [,8]          [,9]
##   [1,]  2.470197e-02  0.0338488586 -7.115866e-02  0.0180861256
##   [2,]  2.000261e-02 -0.1409691955 -6.614998e-01 -0.5752074932
##   [3,] -2.878616e-01  0.1858920371  1.895521e-02 -0.0781030033
##   [4,] -6.549112e-01  0.0061944224  1.430800e-01  0.0288348413
##   [5,]  1.237779e-01 -0.1219398551 -9.728167e-02  0.0567525889
##   [6,]  4.072717e-01 -0.3703207689 -1.457888e+00 -0.0269852311
##   [7,]  2.323455e-01 -0.1097152709 -3.946887e-02  0.0565798352
##   [8,] -1.845269e-01  0.0503276333 -9.787578e-02 -0.0301681354
##   [9,]  5.130038e-01 -0.3504298612  7.593698e-01 -0.0072666810
##  [10,] -1.224262e-01 -0.0256954022  4.651640e-02  0.1034273163
##  [11,]  2.169564e-01 -0.0607369446 -3.204754e-02 -0.0185569686
##  [12,]  1.856443e-01 -0.1392131632  2.410362e-02 -0.1092138133
##  [13,] -6.766145e-01  0.5389941113  4.684409e-01  0.3325237446
##  [14,]  4.458885e-02  0.0536590741  1.716028e-01 -0.1758673272
##  [15,]  1.774845e+00  1.0181771036 -7.404410e-01 -2.0965494893
##  [16,]  3.099177e-01 -0.4016118597  5.791753e-01 -0.3136263891
##  [17,]  1.386883e-01 -0.1976560553  4.792934e-02 -0.0588683739
##  [18,]  1.012646e-01 -0.1068812555 -4.445051e-02 -0.0433590013
##  [19,]  2.367992e-01 -0.5642416917 -8.975761e-02 -0.6013836102
##  [20,] -1.752168e-01  0.1420943971 -2.430758e-01  0.1190331678
##  [21,]  4.099368e-01  0.6766196056  4.924683e-01  1.1794316400
##  [22,]  5.281670e-01  0.0031454467  5.224265e-01 -0.3589826188
##  [23,]  7.292228e-02 -0.0052112694  1.574403e-02 -0.0733415328
##  [24,]  1.523420e-01 -0.0340544029 -3.639761e-02 -0.0170166633
##  [25,]  2.404118e-01  0.2988427866 -2.344282e-01  0.0068187215
##  [26,] -1.235778e-01  0.1400668421  1.868117e-02  0.0874138522
##  [27,]  6.212564e-02 -0.0569259412  2.122119e-02  0.0025767529
##  [28,]  1.856443e-01 -0.1392131632  2.410362e-02 -0.1092138133
##  [29,]  1.595789e-01 -0.0297710716  9.005009e-02  0.1894571078
##  [30,]  1.375367e-01 -0.0318938111  2.009410e-02 -0.0748818381
##  [31,]  1.482207e-01 -0.0484383634 -6.827622e-02 -0.0937044407
##  [32,]  1.085231e-01  0.9296197777  1.341422e-01  0.6171986252
##  [33,] -1.011395e-01 -0.0316873733 -7.742605e-02  0.0076099187
##  [34,] -2.598253e-01  0.0935547273 -1.385553e-02 -0.0098052276
##  [35,]  1.856443e-01 -0.1392131632  2.410362e-02 -0.1092138133
##  [36,] -3.922253e-01  0.1131144846 -1.017815e+00 -0.4284117196
##  [37,] -3.549556e-01  0.2213164887 -4.713437e-02  0.4102515858
##  [38,]  8.955963e-01 -0.0487788315  1.039544e+00 -0.3490868539
##  [39,]  9.266947e-01  0.1510109748  3.944102e-01 -0.1208409381
##  [40,]  2.920435e-01 -0.4220240984 -7.399867e-01 -0.4082993451
##  [41,] -5.197239e-01  0.9419055152 -1.107961e-01  0.7240792741
##  [42,] -2.164333e-01 -0.7357345785 -4.324371e-01  0.1222914965
##  [43,] -2.153651e-02  0.7258773698 -7.615301e-01 -0.2011460415
##  [44,]  1.448362e-01 -0.2469846465 -1.541200e-01 -0.0377938760
##  [45,]  4.197614e-01  1.1173746333 -2.029114e-01 -1.6778528006
##  [46,]  1.897657e-01 -0.1248292027  5.598223e-02 -0.0325260360
##  [47,]  1.237779e-01 -0.1219398551 -9.728167e-02  0.0567525889
##  [48,] -5.630622e-01 -0.0429053682  2.012573e-01  0.3327321873
##  [49,] -9.124823e-03 -0.5485687649  2.443495e+00 -0.4883517085
##  [50,] -5.991900e-01  0.9488266331 -4.267392e-02  0.4904482992
##  [51,] -6.656849e-01 -0.6156955863 -2.298904e-01 -0.2630551254
##  [52,] -9.432540e-01  0.9207454521  4.733810e-01 -0.2020548100
##  [53,] -3.982063e-01  0.1041955796  4.678172e-01 -0.2092421811
##  [54,]  4.594423e-01 -1.5801720223 -6.647786e-01  0.4977352257
##  [55,] -1.293581e+00 -1.0329505828 -1.775642e+00 -0.7834742973
##  [56,]  5.220577e-01  0.9971028403  1.814454e-01  0.5200323017
##  [57,]  1.366079e-01  0.3015905672  5.108780e-01  0.4368675756
##  [58,] -6.969333e-01 -0.1920109524  1.473703e+00 -0.4136594097
##  [59,] -1.022899e+00  0.0560455551  3.402252e-01 -0.7445198275
##  [60,] -1.335945e-01 -0.0245006960  1.515733e-01 -0.0527968792
##  [61,] -9.980526e-01 -0.0824645946  9.288767e-01 -0.0512384843
##  [62,] -8.785688e-01  0.7672634470 -7.652316e-01 -0.1301324933
##  [63,]  1.897657e-01 -0.1248292027  5.598223e-02 -0.0325260360
##  [64,] -1.053958e+00 -0.2697006818  2.544131e-01 -0.7961563102
##  [65,]  1.012646e-01 -0.1068812555 -4.445051e-02 -0.0433590013
##  [66,] -5.628189e-01  0.5731047048 -5.916288e-01 -0.5268406746
##  [67,]  8.272937e-01  1.2965902894 -1.045745e+00 -1.8000002281
##  [68,] -7.029008e-02  0.2076817615 -1.001673e-01  0.0808104424
##  [69,]  1.606133e-02  0.0144548965  8.373686e-02  0.1916629047
##  [70,] -1.587416e+00  0.2087800420 -3.072651e-01  0.0911223637
##  [71,] -5.993706e-01  0.4755119639  9.011372e-02  0.4656140914
##  [72,] -3.744959e-01 -0.6477079711  1.397378e+00 -1.2554701892
##  [73,] -2.839839e-01  0.4002407645 -8.740776e-01  0.8196736994
##  [74,] -9.304336e-01  0.2709012568  2.117622e-01 -0.1534599816
##  [75,]  3.232047e-01  0.1066302193 -2.030714e-02 -0.0245540293
##  [76,] -4.376518e-01  0.2055794432 -4.401769e-02  0.0678312172
##  [77,] -7.225454e-02  0.1732770074  1.313646e-01 -0.2166828241
##  [78,]  4.744079e-02 -0.1230899309 -8.340314e-02 -0.1407794257
##  [79,] -3.642657e-01  0.1295497249  9.806563e-02  0.2610502826
##  [80,] -1.106719e-01 -0.1809050652  3.877952e-02  0.0424459855
##  [81,] -2.364125e-01  0.2058095117 -7.257159e-02  0.1803818159
##  [82,] -2.711003e-01  0.1198360172  2.119595e-01  0.0523600297
##  [83,] -1.313983e-01  2.1686678072 -5.783720e-01 -0.6553609198
##  [84,] -8.669665e-01 -0.6233170191 -1.712377e-01 -0.0003142685
##  [85,] -8.512566e-01  0.4551014205  1.844992e-01  0.3463368216
##  [86,]  1.968316e-01 -0.9717224602 -9.399909e-01  0.7955113067
##  [87,]  1.012646e-01 -0.1068812555 -4.445051e-02 -0.0433590013
##  [88,] -2.019194e-01 -0.1063389408 -9.255296e-02 -0.0394650663
##  [89,]  1.523420e-01 -0.0340544029 -3.639761e-02 -0.0170166633
##  [90,] -1.605603e-01  0.0352170141 -2.185362e-02  0.0608698186
##  [91,]  1.012646e-01 -0.1068812555 -4.445051e-02 -0.0433590013
##  [92,]  1.897657e-01 -0.1248292027  5.598223e-02 -0.0325260360
##  [93,]  1.482207e-01 -0.0484383634 -6.827622e-02 -0.0937044407
##  [94,]  1.523420e-01 -0.0340544029 -3.639761e-02 -0.0170166633
##  [95,] -4.783829e-02 -0.0987848797 -1.349402e-02  0.0249975380
##  [96,]  2.470197e-02  0.0338488586 -7.115866e-02  0.0180861256
##  [97,] -2.740909e+00 -3.1414855107 -1.066917e+00 -0.8478160962
##  [98,]  6.023418e-01 -0.6495243667  4.400255e-03  0.4708812953
##  [99,] -3.933334e-01 -0.1511219911 -1.413549e-01 -0.4576449633
## [100,]  3.854110e-01  0.4537857175 -1.594018e-01 -0.7367636867
## [101,] -2.314829e-01 -0.0081152588 -7.405006e-02  0.0201773040
## [102,] -2.485606e-01  0.4140362043  5.015157e-01 -0.0263043979
## [103,]  5.039347e-01  0.4639539146 -5.611163e-01 -0.5882893231
## [104,] -5.090495e-01  1.5719984541 -1.810639e-01 -0.1415165616
## [105,] -3.035435e-01 -0.0345286345 -2.826018e-01 -0.1337629841
## [106,]  6.233930e-01 -0.3191110970 -1.187688e-02  0.6739149453
## [107,]  2.028041e-01  0.0379528021  2.642787e-01 -0.1181670876
## [108,]  2.103179e-01  1.5894128911 -2.386492e-01 -0.6864756037
## [109,] -3.619941e-01  0.4548002340  2.822758e-01  0.0244663392
## [110,]  6.942872e-01  0.8952470313 -1.141839e+00 -0.2160861581
## [111,]  9.469044e-02  0.4433082124  3.631730e-01  0.4943582792
## [112,] -4.335820e-01  0.1001685496 -4.455675e-01  0.2588494361
## [113,] -6.554194e-01  0.5177973984  4.567600e-02 -0.1892064139
## [114,]  2.130606e-01 -0.2447771343  2.258582e-01 -0.1925122638
## [115,] -6.912524e-01  0.7935702584 -1.562219e-01 -0.0251109264
## [116,]  1.173926e+00 -0.0013165702  2.944144e-01 -0.7426661084
## [117,]  2.497241e-01 -0.2455094171  2.932105e-01 -0.1867895180
## [118,] -5.489761e-01  0.3578526902  1.754228e-02  0.0841926869
## [119,] -4.297657e-01  0.0814625839 -7.514700e-02  0.1478340013
## [120,] -4.832101e-01  0.1654079043  4.972759e-02  0.0986658458
## [121,]  7.113954e-02  0.3518350547  2.435463e-01  0.5139220937
## [122,]  3.710002e-03  0.8539206705  3.024300e-01 -0.1910237578
## [123,]  2.310219e+00 -0.4139129786 -7.388407e-01  0.6342181120
## [124,]  1.897657e-01 -0.1248292027  5.598223e-02 -0.0325260360
## [125,]  1.686182e+00 -0.2951223332  1.371453e+00  1.0381396333
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## [132,] -1.497098e-01  0.2365248951 -4.802570e-02  0.0244855729
## [133,] -6.528125e-02  0.0109119628 -9.176273e-02 -0.1049071451
## [134,] -2.856198e-01  0.2969593701  2.327289e-01 -0.0878257692
## [135,]  1.386883e-01 -0.1976560553  4.792934e-02 -0.0588683739
## [136,]  8.879983e-02  0.0184660240 -1.270379e-02 -0.0404445408
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## [620,]  2.222389e-01  0.2499254460 -8.899317e-02 -0.0324725341
## [621,] -1.795476e-01  0.1344566541 -8.464341e-01 -0.2464932762
## [622,] -2.898340e-01  0.2910339270  1.298689e-02  0.1912400700
## [623,]  1.545658e-01 -0.1739787619  1.948152e-02 -0.0259713819
## [624,]  1.005165e-01 -0.0483072474 -6.005780e-02  0.1355853350
## [625,]  1.770792e-01 -0.1890373615 -3.334965e-02  0.0741402082
## [626,]  7.292228e-02 -0.0052112694  1.574403e-02 -0.0733415328
## [627,]  7.292228e-02 -0.0052112694  1.574403e-02 -0.0733415328
## [628,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [629,]  2.015219e-01 -0.1155358698 -4.344195e-03 -0.0763168213
## [630,]  7.292228e-02 -0.0052112694  1.574403e-02 -0.0733415328
## [631,] -3.882188e-01  0.1695747405 -3.478081e-03  0.2262029515
## [632,]  2.281566e-01 -0.1162105089 -2.529676e-02  0.1004825461
## [633,] -8.283834e-01 -0.3245099982  8.293664e-02 -0.0619566567
## [634,]  7.292228e-02 -0.0052112694  1.574403e-02 -0.0733415328
## [635,] -5.050149e-01  0.2307675541 -1.797061e-01  0.1118084053
## [636,] -4.883546e-02  0.0320728509  1.738128e-03  0.2298812444
## [637,] -1.605064e-01  0.1848102232 -4.254854e-02  0.1004038586
## [638,]  1.386883e-01 -0.1976560553  4.792934e-02 -0.0588683739
## [639,]  3.549861e-02  0.0855635304 -7.663582e-02 -0.0578321601
## [640,]  7.292228e-02 -0.0052112694  1.574403e-02 -0.0733415328
## [641,]  6.212564e-02 -0.0569259412  2.122119e-02  0.0025767529
## [642,]  4.907363e-02  0.0822958605 -2.385818e-01 -0.4025322010
## [643,]  5.348254e-01  0.5080443414  8.499206e-01  1.0942907442
## [644,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [645,]  1.897657e-01 -0.1248292027  5.598223e-02 -0.0325260360
## [646,]  1.012646e-01 -0.1068812555 -4.445051e-02 -0.0433590013
## [647,]  1.437014e-01 -0.0534483649  1.184979e-01  0.1565601158
## [648,]  1.437014e-01 -0.0534483649  1.184979e-01  0.1565601158
## [649,]  9.543563e-02 -0.0202698689 -3.708714e-02  0.0267700573
## [650,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [651,] -8.101485e-01  0.1320735482 -2.574912e-01  0.3573522599
## [652,]  3.549861e-02  0.0855635304 -7.663582e-02 -0.0578321601
## [653,]  4.348448e-01  0.7257498959  3.254585e-01  0.2229440420
## [654,] -2.556194e-01  0.3427805307 -1.017638e-01  0.3241032639
## [655,] -8.279263e-01  0.9545507422 -7.582807e-02  0.9801873291
## [656,] -4.749760e-01  0.2742751737 -8.719858e-02  0.2057778895
## [657,]  1.482207e-01 -0.0484383634 -6.827622e-02 -0.0937044407
## [658,] -6.433652e-01  0.3273949753 -2.004220e-01  0.0313845427
## [659,]  1.897657e-01 -0.1248292027  5.598223e-02 -0.0325260360
## [660,]  2.880049e-01  0.0574806601 -5.680785e-02 -0.0179993753
## [661,] -1.429819e-01 -0.0260632060  2.638268e-02  0.0310102693
## [662,]  7.800319e-02 -0.0332486479 -7.226630e-03  0.0354737449
## [663,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [664,]  2.015219e-01 -0.1155358698 -4.344195e-03 -0.0763168213
## [665,] -2.608560e-02  0.5519345549 -1.535858e-01 -0.0419155803
## [666,]  6.716200e-01 -0.0870026486  1.032482e-01  0.5661955997
## [667,] -1.979301e-01  0.2755850230 -1.349284e-01  0.1159132312
## [668,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [669,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [670,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [671,]  2.056432e-01 -0.1011519093  2.753441e-02  0.0003709560
## [672,]  8.596067e-02  0.1575707355  2.240405e-01 -0.1589825844
## [673,]  1.545658e-01 -0.1739787619  1.948152e-02 -0.0259713819
## [674,]  6.962905e-01  0.2968585377 -9.284018e-02 -0.4049759434
## [675,]  2.281566e-01 -0.1162105089 -2.529676e-02  0.1004825461
## [676,] -5.067049e-01 -0.3488160511  4.675565e-01  0.0400789207
## [677,]  8.879983e-02  0.0184660240 -1.270379e-02 -0.0404445408
## [678,]  2.799277e-02 -0.1273667132 -1.943582e-01 -0.0786093728
## [679,] -3.727639e-01  0.1349177629 -2.461952e-02 -0.0922809963
## [680,]  2.015219e-01 -0.1155358698 -4.344195e-03 -0.0763168213
## [681,] -1.049028e-01  0.0388431973 -1.056584e+00 -0.4600302013
## [682,] -2.810270e-01 -0.7339208773 -7.165207e-02 -0.3103001085
## [683,] -1.710879e-01 -0.6988562830  1.203811e+00 -0.1913936636
##               [,10]         [,11]         [,12]         [,13]
##   [1,]  0.081313250 -0.0491149789  0.0698191540 -0.0434645550
##   [2,] -0.013585072  0.3604850878  0.6350158794 -0.6041697264
##   [3,] -0.001202664  0.1678785340 -0.0049387548  0.1986346895
##   [4,] -0.752041025 -1.6780059840 -0.5186168204  0.0020604523
##   [5,] -0.256693193  0.1221129915  0.0798966032 -0.2121968839
##   [6,] -1.132133147  0.4671433783  0.2551311233 -0.5515074425
##   [7,] -0.038651998 -0.0652520405 -0.1018106065  0.0123542214
##   [8,]  0.194657571  0.0443664888 -0.1144553556  0.0355697288
##   [9,]  0.456566280 -0.1053210567 -0.8353562985 -0.6836118978
##  [10,]  0.141146530  0.0912547737 -0.0050672392  0.0810184972
##  [11,] -0.144232062  0.0456836073 -0.0596699756 -0.0655780035
##  [12,]  0.024645234  0.0538077760 -0.0649080294 -0.0551621917
##  [13,]  0.478994483  0.1417004910  0.4821832081 -0.5002567575
##  [14,]  0.098570142  0.0644263450 -0.2478161029 -0.3093325937
##  [15,]  0.075333881  1.0925032792 -0.3214883403 -0.3177334569
##  [16,]  0.299385433  0.8175363109  0.6812337157  0.8251760079
##  [17,] -0.066204209  0.0007120358  0.0199733564  0.0193135164
##  [18,] -0.095022399 -0.0240125173 -0.0063444072 -0.0592129424
##  [19,]  0.509805613 -0.8039813188  0.4664670923  0.6852912101
##  [20,] -0.068049790 -0.1942684800 -0.0272419108 -0.1351291366
##  [21,]  1.482876433  0.2657808614 -0.4667715694  1.1172638555
##  [22,] -0.440657443  1.1070839717  0.7269025753 -1.2941401303
##  [23,] -0.026931145  0.1078964081  0.0068333475  0.1201497387
##  [24,] -0.066708821  0.0143091919 -0.0401337428 -0.0230676680
##  [25,]  0.671817480 -0.1911467982  0.5656596293 -1.0895859644
##  [26,]  0.102896352  0.2298135614 -0.0377434809  0.1393443841
##  [27,]  0.110131440 -0.0243904258  0.0961369176  0.0350619037
##  [28,]  0.024645234  0.0538077760 -0.0649080294 -0.0551621917
##  [29,]  0.117251049 -0.1325797156 -0.0538090993 -0.2186117776
##  [30,] -0.104454387  0.1392708235 -0.0127028853  0.0776394032
##  [31,] -0.004172956  0.0290832229 -0.0912257930 -0.1336886505
##  [32,] -0.898488563 -0.5944561453  0.7831445832 -0.0564312399
##  [33,] -0.061335511 -0.0026816505 -0.1381243203  0.0361590682
##  [34,]  0.171899382  0.1231796740 -0.0163962152  0.2894081180
##  [35,]  0.024645234  0.0538077760 -0.0649080294 -0.0551621917
##  [36,]  0.931141843  0.4892266368 -0.3567856025 -0.0446297648
##  [37,] -0.013926767 -0.0095423956 -0.0205603558 -0.2198523917
##  [38,]  0.844845676 -0.3908647065 -0.2280890844 -0.3952148477
##  [39,]  0.132070623  1.0804818174 -0.2158769192 -2.0174275207
##  [40,]  0.846794664 -0.2104538577 -0.7152588209 -1.1450998084
##  [41,] -1.237569544 -0.5724973444 -0.2527346852  0.1972712301
##  [42,] -0.432466478  0.0906012253 -0.5415362474  0.1007654171
##  [43,]  0.091182090 -0.7551564636  0.2988902817 -0.0590475791
##  [44,]  0.207813828 -0.0378432627 -0.0026561493  0.1398303111
##  [45,] -0.069951872  0.2944233534  0.6083900126  0.8830940748
##  [46,] -0.037890631  0.0390337450 -0.0138159792  0.0554587908
##  [47,] -0.256693193  0.1221129915  0.0798966032 -0.2121968839
##  [48,]  0.815020579 -0.2484997442 -0.5617110591  1.2435858185
##  [49,]  0.112714760  0.3075331485 -1.5053852489 -0.5861517490
##  [50,]  0.463834368 -0.6783350431  0.2891652873 -0.5078068824
##  [51,]  0.013299997 -0.3909273236  1.1105396027  0.4797325127
##  [52,] -1.278171930  1.0597588562  0.1014728879  0.2881869750
##  [53,] -1.061018768 -0.1783420431 -0.0898770699 -0.0009620974
##  [54,] -0.544646601 -0.5472419574  0.6181015811  0.0651901521
##  [55,] -0.109571000  1.1269652204 -1.2447105834 -0.3827041722
##  [56,]  1.134322907  0.6365291764  0.0349797384  0.4881664401
##  [57,]  0.757948750 -0.0721544159 -0.1140703788 -0.0745733839
##  [58,] -0.644631146  0.6846294275 -0.3163965726  0.5604271661
##  [59,] -0.193200034 -0.1494159005  0.8791604644 -0.7965536382
##  [60,]  0.016656040  0.1237404240  0.0007296821  0.2124702003
##  [61,] -0.467093082  1.0636012555 -0.3154509002  0.5284402964
##  [62,]  0.599922855  0.3276911032 -0.4415759279  0.0128677554
##  [63,] -0.037890631  0.0390337450 -0.0138159792  0.0554587908
##  [64,]  0.047269893  0.7325975033  0.6064411316  1.3389391442
##  [65,] -0.095022399 -0.0240125173 -0.0063444072 -0.0592129424
##  [66,]  0.033196348 -0.0872269623 -0.1884716476  0.2813301683
##  [67,] -0.547658086  0.6356332510 -1.1669705195  0.4121390983
##  [68,] -0.070654274  0.0150013647 -0.0299269256  0.0217715698
##  [69,]  0.303946853 -0.1252266336  0.0146250200 -0.1764943328
##  [70,]  0.149671713  0.7629822730 -0.0510816024  1.0007124900
##  [71,]  0.064480822 -0.1529984612 -0.1235493897 -0.2327384989
##  [72,]  0.156199482  0.8573654344  0.2483314642  0.4388158745
##  [73,] -0.851192619 -0.3629344605 -0.6785631811 -0.1388902211
##  [74,]  0.145633620  0.2056915519  0.3826575762  0.6210443360
##  [75,]  0.007660224  0.2225192561 -0.1576904412  0.0032051512
##  [76,]  0.127155596 -0.1543629750  0.0327679677  0.0960259721
##  [77,]  0.109529628  0.1332890081 -0.2271667762 -0.2446416458
##  [78,] -0.066189757 -0.0731812579 -0.1966051372 -0.1730846413
##  [79,]  0.248780595  0.2290925732 -0.1077738006 -0.0491535264
##  [80,] -0.123366765 -0.0310528376 -0.0269251710  0.1891612351
##  [81,]  0.288663989  0.0414277590  0.0447785584  0.0182404259
##  [82,] -0.076907138  0.2306874650  0.0463196286  0.0947167890
##  [83,] -1.359475431  0.5920886614  0.0869193411 -0.0489563449
##  [84,] -0.031889942  0.2519424406  0.9725064782  0.9292072462
##  [85,]  0.409590238  0.3491155362  1.3775889744  0.5998315131
##  [86,] -0.133744295 -0.2116331104 -0.4369003866  0.4740650737
##  [87,] -0.095022399 -0.0240125173 -0.0063444072 -0.0592129424
##  [88,] -0.123352312 -0.1049461313 -0.2435036645 -0.0032369226
##  [89,] -0.066708821  0.0143091919 -0.0401337428 -0.0230676680
##  [90,] -0.122767433  0.0053542819  0.0014535977  0.2274831255
##  [91,] -0.095022399 -0.0240125173 -0.0063444072 -0.0592129424
##  [92,] -0.037890631  0.0390337450 -0.0138159792  0.0554587908
##  [93,] -0.004172956  0.0290832229 -0.0912257930 -0.1336886505
##  [94,] -0.066708821  0.0143091919 -0.0401337428 -0.0230676680
##  [95,] -0.071191054 -0.0487343502 -0.0702877792  0.0521711951
##  [96,]  0.081313250 -0.0491149789  0.0698191540 -0.0434645550
##  [97,] -0.320031618  2.5045530458 -2.1142792199  0.2685792820
##  [98,]  1.005926621  1.0131984333 -0.8905559653  1.0465674737
##  [99,] -0.225181037 -0.0807772424  0.3989252783 -0.0117775059
## [100,]  0.214483171 -0.0670263048 -0.2578141225 -0.8728935044
## [101,]  0.103808128 -0.0087292514 -0.0295739699  0.1100454369
## [102,]  0.647701606  0.4235392724 -0.2382766553 -1.0326723413
## [103,] -0.096375386  0.2764936403 -1.1095397484  1.0178671914
## [104,]  0.116869197  0.0593016081 -0.8681174349 -0.4434278859
## [105,] -0.003530423  0.1375394953  0.5151452214  0.2663700274
## [106,] -1.349291551 -1.1897887376 -0.5604586788  0.6585420051
## [107,] -0.316440332  0.2514021391 -0.0117935389  0.3302824997
## [108,] -1.088715897  0.5826368138 -0.7821217000 -0.0447614212
## [109,] -0.013446064 -0.0266245811 -0.0660391534  0.8821246204
## [110,] -0.588253376  1.2437843506 -0.1887182020 -0.3348985371
## [111,] -0.448836353 -0.2384488494 -0.4551755912  0.0788095574
## [112,]  0.086102967  0.3635743175 -0.0298424686  0.2661064563
## [113,]  0.382546867  0.0503645982 -0.2668667613 -0.1842669169
## [114,] -0.208375564 -0.1855634177  0.3402098126 -0.0884943826
## [115,]  0.242688953  0.0092870721 -0.0231342435 -0.3665264550
## [116,] -1.000276739  0.3252226312 -1.1628795343 -0.2434164251
## [117,]  0.081086957 -0.2217348207 -0.1403330081 -0.1917016432
## [118,]  0.206148075  0.2584212719 -0.0299793504  0.2603396704
## [119,]  0.074959151 -0.0021724156 -0.1102618328  0.3160384300
## [120,]  0.166875011  0.1512368997 -0.0168393415  0.1595034480
## [121,] -0.827897124  0.0123518971 -0.5582762851  1.0442689497
## [122,] -0.816067138 -0.4921928662  0.1348388238  0.4151280896
## [123,]  1.784825637  1.5623717739  3.3523732400 -0.1172165410
## [124,] -0.037890631  0.0390337450 -0.0138159792  0.0554587908
## [125,]  0.691815275  0.3113817309 -1.6840662453 -0.9465711722
## [126,] -0.055749335  0.0831718550 -0.0194844162  0.0416232800
## [127,]  0.272481883 -0.5989528226  0.5416785693 -0.5625463508
## [128,] -0.225743590  0.1742796096 -0.2305783881  0.2330087000
## [129,]  0.303946853 -0.1252266336  0.0146250200 -0.1764943328
## [130,] -0.004172956  0.0290832229 -0.0912257930 -0.1336886505
## [131,] -0.822433374 -1.5539804694 -0.2727453203  0.0894831672
## [132,] -0.030876599  0.1085885810  0.0170401647  0.1649889764
## [133,] -0.117766136 -0.0190926258 -0.1248637604  0.0022272891
## [134,] -0.142689780  0.2479600942  0.0863872557  0.3883705634
## [135,] -0.066204209  0.0007120358  0.0199733564  0.0193135164
## [136,] -0.065604878  0.0371191553  0.0483521250  0.0576354068
## [137,]  0.132626318  0.0159953017 -0.0032562062  0.1885718956
## [138,] -0.065604878  0.0371191553  0.0483521250  0.0576354068
## [139,] -0.014028499 -0.0169694768 -0.0233892518 -0.1176765237
## [140,] -1.437563744  0.4320475207  0.3634367014  0.4420197253
## [141,] -0.208375564 -0.1855634177  0.3402098126 -0.0884943826
## [142,]  0.024645234  0.0538077760 -0.0649080294 -0.0551621917
## [143,] -0.039451915  0.7674747829  0.8936242405  0.3168397302
## [144,] -0.074178950 -0.0032486098 -0.1309674257  0.0945477507
## [145,] -0.590783107 -0.2733698392 -0.0113848405  0.5472542468
## [146,]  0.020981083 -0.2852916290 -0.5571450813  0.3944779607
## [147,] -0.144232062  0.0456836073 -0.0596699756 -0.0655780035
## [148,]  0.420755258  0.5346958562 -0.6196239164  0.3722817804
## [149,]  0.190675846 -1.2349851752  0.2387852242  0.2060286939
## [150,]  0.060401021 -0.0199480639 -0.1461902261 -0.3294657413
## [151,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [152,] -0.420702904 -0.2195866712 -0.4191692592 -0.2339394456
## [153,]  0.368290635  0.1448597488 -0.0620861375  0.2864221509
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## [611,] -2.014973133  0.8774292705 -0.2842614195 -0.5634267115
## [612,] -0.145839699 -0.1707893867  0.2891177625 -0.1991153651
## [613,] -0.014028499 -0.0169694768 -0.0233892518 -0.1176765237
## [614,] -0.104877942 -0.0700652170  0.0614921339 -0.0432008156
## [615,]  0.323260819 -0.2506146495 -0.0659578629 -0.2689682654
## [616,]  0.110131440 -0.0243904258  0.0961369176  0.0350619037
## [617,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [618,]  0.603964988  0.0623050320  0.5531299852 -0.3720672107
## [619,] -0.065604878  0.0371191553  0.0483521250  0.0576354068
## [620,] -0.020054023  0.2206046664 -0.0955223370  0.0053817672
## [621,] -0.384951424 -0.2480846342  0.4618288244 -0.4688588353
## [622,]  0.079573262  0.5055724727 -0.0481550295  0.0327923257
## [623,] -0.104877942 -0.0700652170  0.0614921339 -0.0432008156
## [624,] -0.090213087  0.0509578302  0.2238967055 -0.1804363697
## [625,] -0.266548736  0.0760602918  0.1477331443 -0.1961847570
## [626,] -0.026931145  0.1078964081  0.0068333475  0.1201497387
## [627,] -0.026931145  0.1078964081  0.0068333475  0.1201497387
## [628,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [629,] -0.014028499 -0.0169694768 -0.0233892518 -0.1176765237
## [630,] -0.026931145  0.1078964081  0.0068333475  0.1201497387
## [631,]  0.329616902  0.0740824960 -0.0205673599  0.2239078190
## [632,] -0.238235158  0.1143820010  0.1139438087 -0.1600394826
## [633,]  0.369679177  0.0697053691  0.1422900403  0.5767866570
## [634,] -0.026931145  0.1078964081  0.0068333475  0.1201497387
## [635,] -0.345151417 -0.5151803117 -0.0983979155 -0.0130134366
## [636,]  0.007789314  0.2757019917  0.0473844356 -0.0358201698
## [637,]  0.106185986 -0.0236982529  0.1063437348  0.0799011415
## [638,] -0.066204209  0.0007120358  0.0199733564  0.0193135164
## [639,] -0.055749335  0.0831718550 -0.0194844162  0.0416232800
## [640,] -0.026931145  0.1078964081  0.0068333475  0.1201497387
## [641,]  0.110131440 -0.0243904258  0.0961369176  0.0350619037
## [642,] -0.490163786 -0.6804785660  0.5594608488 -0.2324187003
## [643,] -0.194020539 -0.5159915095 -0.9208439035  0.5829939323
## [644,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [645,] -0.037890631  0.0390337450 -0.0138159792  0.0554587908
## [646,] -0.095022399 -0.0240125173 -0.0063444072 -0.0592129424
## [647,]  0.155924782 -0.0618024628 -0.0953278769 -0.1560974457
## [648,]  0.155924782 -0.0618024628 -0.0953278769 -0.1560974457
## [649,] -0.188601939  0.2540219169  0.0930743578 -0.0328342027
## [650,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [651,]  0.666180877 -0.0979835558  0.0536468395  0.4577301710
## [652,] -0.055749335  0.0831718550 -0.0194844162  0.0416232800
## [653,]  0.063049551  0.2156964798  0.3336357956 -0.4947347690
## [654,]  0.202219218  0.1043490141  1.0627011283 -0.4510901215
## [655,]  0.336233064  0.5878631724  0.8243433256  0.1037068561
## [656,]  0.094438849  0.1419497195 -0.0914235391  0.2731759796
## [657,] -0.004172956  0.0290832229 -0.0912257930 -0.1336886505
## [658,]  0.253639955 -0.3050811223 -0.0337413978 -0.0850001489
## [659,] -0.037890631  0.0390337450 -0.0138159792  0.0554587908
## [660,] -0.059327087  0.1134202941 -0.0823823281 -0.0954544552
## [661,]  0.160939896  0.0543170109 -0.0370455419  0.2247171700
## [662,]  0.071457707 -0.0951676786  0.1376556952 -0.0274524282
## [663,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [664,] -0.014028499 -0.0169694768 -0.0233892518 -0.1176765237
## [665,]  0.117332235 -0.7016895645  0.2716584840 -0.1609000226
## [666,] -0.104220255 -1.1148788130 -0.1516786535  0.2509384189
## [667,]  0.077367796 -0.0484228060  0.0800259712  0.0013746827
## [668,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [669,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [670,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [671,] -0.076564364 -0.0317435078  0.0277027983 -0.0070555411
## [672,] -0.305480846  0.3202648022  0.0088557878  0.3949734476
## [673,] -0.104877942 -0.0700652170  0.0614921339 -0.0432008156
## [674,] -0.597089752  0.2562081676 -0.5677921427  0.2988248151
## [675,] -0.238235158  0.1143820010  0.1139438087 -0.1600394826
## [676,] -0.030785964 -0.3706141429  1.4376873786 -0.1111169726
## [677,] -0.065604878  0.0371191553  0.0483521250  0.0576354068
## [678,]  0.218773314  0.0310194004  0.0179931774  0.2045212590
## [679,] -0.101893198 -0.0051631995 -0.0687993215  0.0967243667
## [680,] -0.014028499 -0.0169694768 -0.0233892518 -0.1176765237
## [681,]  0.747839423  0.2167064085  0.3355710551 -0.8826284897
## [682,]  0.090629136 -1.8753502109  0.2180475440 -0.1324892066
## [683,] -0.116657891 -1.6008483031  0.6481171382 -0.0154397050
##                [,14]        [,15]        [,16]        [,17]         [,18]
##   [1,] -1.389636e-01  0.146484450  0.152780195  0.194313810  0.0045938050
##   [2,]  4.588588e-02  0.183284014  0.110494526 -0.539450983  0.3473595954
##   [3,] -8.891983e-02  0.037449764 -0.154037553  0.044754210  0.2902846840
##   [4,]  4.807221e-02  0.536178956 -0.074295799  0.012890010 -0.8885848353
##   [5,] -1.235015e-01  0.020277487  0.041921315 -0.424490630 -0.1466515048
##   [6,] -1.205792e+00 -0.436982124  0.084167801 -0.231444184 -0.3582352201
##   [7,]  9.228478e-02  0.263316491 -0.023152410  0.190120145  0.0883015898
##   [8,] -1.452565e-01 -0.289174719  0.056016470  0.174375517  0.0802790351
##   [9,] -6.886671e-01 -0.009842556  0.206759837  0.196786331  0.0541596439
##  [10,] -1.470249e-01 -0.457157666  0.146343444 -0.220915542  0.0547708350
##  [11,]  2.538375e-01  0.278734157 -0.027435750  0.003256980 -0.0347897982
##  [12,]  7.354245e-02 -0.014564148 -0.031559937  0.207406520  0.1126730679
##  [13,] -4.916818e-01  0.043534421 -0.102599221 -0.972320505 -1.0900903284
##  [14,]  9.403964e-02  0.258676023  0.115337240 -0.391297012 -0.1733574426
##  [15,] -3.155707e-01 -1.820939818 -1.105185591  0.174042823 -0.4472031596
##  [16,]  6.011780e-01  0.931988582  0.515576407 -0.355808780 -0.8966646929
##  [17,] -7.610927e-03 -0.046071389  0.099728759 -0.137436567  0.1192117011
##  [18,] -2.903096e-02  0.090226960  0.141171352 -0.099988254  0.0790742941
##  [19,] -2.793567e-01 -0.380895127 -0.868655142  0.066304481  0.7310071680
##  [20,]  1.840689e-01  0.362725921 -0.255886104 -0.158870079  0.1195451495
##  [21,]  2.424412e-01 -0.759436535 -0.254145962 -0.005538412 -0.9342486795
##  [22,] -3.354510e-01 -0.828701617 -0.028330332  0.270245828 -0.4773089644
##  [23,]  6.694806e-03 -0.013567502  0.059840962 -0.025602561  0.0484249451
##  [24,]  1.148030e-01  0.228897610 -0.022111758  0.101811047  0.0291131318
##  [25,]  9.384678e-02 -0.889761689  1.104588464  0.506657643 -0.1166833156
##  [26,]  6.315381e-03 -0.374817233  0.101131654 -0.207635602 -0.0799188509
##  [27,] -1.175436e-01  0.010186101  0.111337602  0.156865497  0.0447312119
##  [28,]  7.354245e-02 -0.014564148 -0.031559937  0.207406520  0.1126730679
##  [29,]  1.555524e-01 -0.035564472 -0.304441272 -0.001087270 -0.1125755567
##  [30,]  1.457293e-01  0.036269044  0.054516969 -0.124156628 -0.0154779849
##  [31,]  5.212242e-02  0.121734200  0.009882656  0.244854833  0.0725356610
##  [32,]  6.945418e-01 -0.810021884 -0.458250982  0.304915207 -0.6414630534
##  [33,] -3.449412e-02 -0.001891250 -0.079860604  0.362931511  0.1215241080
##  [34,] -1.906841e-01 -0.424476421  0.105974775 -0.096081878  0.0561683192
##  [35,]  7.354245e-02 -0.014564148 -0.031559937  0.207406520  0.1126730679
##  [36,] -5.064628e-01  0.660670792 -0.857941198  0.361419548 -1.0366257980
##  [37,]  3.104549e-01 -0.290564601 -0.177546504 -0.322076401  0.0343269461
##  [38,] -6.853034e-01  1.671039869  0.865175684  0.841226150 -0.3814731120
##  [39,] -1.073164e-01 -1.320487286 -0.275191288  0.128133502 -0.2082300314
##  [40,]  4.060311e-01  0.306241657 -1.318583141 -0.876885548  0.9743052287
##  [41,]  6.249414e-01 -0.106124994 -0.434603365  0.333597836 -0.0419217260
##  [42,] -8.580561e-01  0.100630241  0.665680633  0.331178028  0.5784058039
##  [43,]  3.617174e-01  0.201866663  0.061201318  0.386485772 -0.5731552343
##  [44,]  2.420261e-01  0.059959446 -0.450664208  0.011291099  0.4073514323
##  [45,]  2.116728e-01 -0.399695603  0.435312831 -0.448908643  0.3966020138
##  [46,]  1.362230e-01  0.092599262 -0.063554351  0.064362733  0.0692505387
##  [47,] -1.235015e-01  0.020277487  0.041921315 -0.424490630 -0.1466515048
##  [48,]  8.040325e-01  0.230335759  0.116098633 -1.093754799  0.7887181279
##  [49,] -1.437313e+00 -0.105918204  0.380562632 -0.209690420  0.1442294250
##  [50,]  3.744800e-02  0.214754427  0.403297444  0.261765509 -1.0223338107
##  [51,]  1.117684e-01  0.100062018 -0.534506934 -0.227956234 -0.2979198858
##  [52,] -4.524497e-01 -0.017641768  0.985049609 -0.227888127  0.4480603112
##  [53,] -6.405663e-01 -0.436119514 -0.087178458  0.498321841  0.0927468708
##  [54,] -3.226263e-01 -0.038013460  0.467246713 -0.605076146 -0.6125228384
##  [55,]  4.658681e-01  0.041219203 -0.301514821  1.078272622 -0.3888274636
##  [56,]  6.364206e-02 -0.393415258 -0.128259667  0.397094934 -0.7772664368
##  [57,] -2.213072e-01 -0.432497999 -0.475936236  0.479991182 -0.5066608402
##  [58,] -5.559846e-01 -0.429780220 -0.708423753  0.886260121  0.5039843961
##  [59,] -6.482053e-01  0.470901415  0.018426020 -0.673513944  0.0333160826
##  [60,]  6.202843e-02  0.007318179 -0.318875458  0.097271191  0.3512476846
##  [61,] -1.503755e-01  0.835989136 -0.021109378 -0.023929661  0.7939436568
##  [62,]  9.303434e-02  0.969209701 -0.696120750  0.335876998  0.3303410470
##  [63,]  1.362230e-01  0.092599262 -0.063554351  0.064362733  0.0692505387
##  [64,]  3.523912e-01  0.168897154 -0.375289835  0.061609713  0.0839657972
##  [65,] -2.903096e-02  0.090226960  0.141171352 -0.099988254  0.0790742941
##  [66,] -6.194412e-03  0.819976671 -0.155874529  0.375881281 -0.7311506504
##  [67,] -1.497028e+00  0.192369298 -1.182909100 -0.521088330 -0.7231669934
##  [68,]  1.157477e-01  0.167075100  0.170287553  0.065192628  0.2911934036
##  [69,] -1.050511e-01 -0.169023995 -0.084864086  0.089788650  0.0234323819
##  [70,]  2.755159e-01 -0.535371718  0.430462841 -0.404616836  0.8703199374
##  [71,]  5.193141e-02 -0.425767797 -0.100465705  0.175016714  0.3161093253
##  [72,]  2.027111e-01  0.465140639 -0.199778464 -0.198665541  0.3181540567
##  [73,]  3.787412e-01  0.691765823  0.576168698  0.248757892 -0.4382111027
##  [74,] -2.128459e-01 -0.241502091 -0.149165947 -0.216793874  0.3592701027
##  [75,]  2.032153e-01  0.377778473  0.118343408 -0.012795023  0.0880952526
##  [76,] -3.206179e-01 -0.111405909 -0.000262874  0.314939710  0.0478203711
##  [77,] -3.548859e-02  0.152509259  0.238732552 -0.481262306 -0.1941830362
##  [78,]  1.452255e-01  0.087078803  0.002216614  0.225626195  0.1113560153
##  [79,] -1.887044e-02 -0.942465241  0.134356070  0.011169195 -0.0049391683
##  [80,] -9.422746e-02 -0.169696840  0.009985499 -0.019359890  0.1682001480
##  [81,] -2.783776e-01 -0.264601827  0.199394880  0.110834836 -0.0598470610
##  [82,]  3.784567e-02 -0.254591027 -0.116988645  0.063580930  0.3334102766
##  [83,] -5.471958e-03 -0.645277432  0.970610603 -0.538065045 -0.4547259418
##  [84,] -2.060678e-02  0.583482046  0.707919682 -0.173583494 -0.7970955880
##  [85,] -2.886914e-01  0.762499350  0.654914134  0.058489965 -0.7078049151
##  [86,] -3.905025e-02  0.260647770  0.538842607 -0.076411396  0.9052380816
##  [87,] -2.903096e-02  0.090226960  0.141171352 -0.099988254  0.0790742941
##  [88,]  5.860895e-02 -0.036546647 -0.087526646  0.343702873  0.1603444622
##  [89,]  1.148030e-01  0.228897610 -0.022111758  0.101811047  0.0291131318
##  [90,] -7.308494e-02 -0.086146590 -0.074587531  0.094100960 -0.0631138733
##  [91,] -2.903096e-02  0.090226960  0.141171352 -0.099988254  0.0790742941
##  [92,]  1.362230e-01  0.092599262 -0.063554351  0.064362733  0.0692505387
##  [93,]  5.212242e-02  0.121734200  0.009882656  0.244854833  0.0725356610
##  [94,]  1.148030e-01  0.228897610 -0.022111758  0.101811047  0.0291131318
##  [95,] -6.237293e-03 -0.087143237 -0.165988429  0.327110040  0.0011342495
##  [96,] -1.389636e-01  0.146484450  0.152780195  0.194313810  0.0045938050
##  [97,]  1.001581e+00 -0.267797852  0.768630043  2.036825089 -0.2058467931
##  [98,]  2.991348e-01 -0.366506946 -0.097886957  0.062514506 -0.9838412871
##  [99,]  4.722510e-02  0.017907679 -0.438440176  0.009268474  0.6556763423
## [100,]  4.973805e-01 -0.214412776  0.248600967  0.117006641  0.2896237833
## [101,] -2.264099e-01 -0.320681960  0.187305166 -0.170467571  0.0868176682
## [102,] -1.145347e-01 -0.475792461 -0.015766599 -0.371806729 -0.3722713390
## [103,]  1.488834e-02 -0.151151909 -0.457666271 -0.539852286  0.3257721386
## [104,]  1.219390e+00  0.406278449  1.888893806 -1.310937965  0.0680155608
## [105,] -4.382386e-01  0.202566175 -0.139233860 -0.051296814 -0.3866231139
## [106,] -5.665792e-02 -0.180178067 -0.273386344  0.889688856 -0.7537953395
## [107,]  2.064607e-01  0.249694509 -0.165714810  0.139983430  0.0768460649
## [108,]  8.132034e-01 -0.131183648  0.801638453 -0.216965260  0.2198988776
## [109,]  1.957443e-01 -0.035271077 -1.212142938  0.626375155  0.4933184346
## [110,] -5.520247e-01  0.576920548  0.569224056 -0.071780238  0.0733706338
## [111,]  5.802674e-01 -0.329825970 -0.485603239  0.427998393 -0.1341169224
## [112,] -3.013287e-01  0.560076657 -0.425050963 -0.435903906 -0.0053823708
## [113,] -3.493253e-01 -0.256366220  0.263009150 -0.511744087 -0.2323389499
## [114,]  3.445770e-01 -0.061982008 -0.229761671 -0.144620344 -0.0877596161
## [115,] -1.557572e-01 -0.106160030 -0.177408706  0.185349439  0.6678186236
## [116,] -9.802699e-01 -1.218153220 -0.658893628 -0.406905940  0.4585182645
## [117,]  1.596573e-02  0.051909331 -0.044221713 -0.745059511 -0.2317583686
## [118,] -2.283338e-01 -0.373636513 -0.107422867 -0.038724765  0.2258438180
## [119,] -1.691925e-01 -0.305636760 -0.065721205  0.149408408  0.0858449529
## [120,] -2.426395e-01 -0.406140400 -0.067535070 -0.150558771  0.2966305740
## [121,]  4.968981e-01 -0.028670807 -0.477876018  0.757219382  0.1627363519
## [122,] -1.303846e-01  0.025836064 -0.323410681  1.003106776 -0.3118292494
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## [682,]  3.791833e-01  0.452674907  1.708102241 -0.132724750 -0.9775073101
## [683,] -2.109338e-01 -0.110926147  0.306401028 -1.218259582  0.0149275941
##                [,19]        [,20]         [,21]         [,22]        [,23]
##   [1,]  0.0065957932  0.102489525 -0.1944101653 -0.2454251045  0.213308960
##   [2,] -0.6481142146  0.466418841 -1.5336294249 -0.1410551051 -0.390591943
##   [3,]  0.0451531393 -0.088541161 -0.1890030445 -0.3786000907  0.006006025
##   [4,]  0.0448789870 -0.679753535 -0.2427409843 -0.2883130374 -0.643338032
##   [5,] -0.1989001318  0.235235982  0.1064292461 -0.3960387477  0.190720633
##   [6,] -0.7388873085  0.233952731 -0.3014758300  0.3312743027 -0.649763530
##   [7,]  0.0203671839 -0.119947910 -0.0955084464 -0.1371023327  0.180632498
##   [8,]  0.0718893897  0.062125714 -0.1598940678 -0.2361098432 -0.098112102
##   [9,]  0.2109169971 -0.067027566 -0.6616505446  0.2111759927  0.303073286
##  [10,] -0.6678734548  0.021853761  0.1283799279  0.0490416287 -0.336375028
##  [11,]  0.0681679715 -0.139230559 -0.0006817074  0.1444085403  0.315162433
##  [12,]  0.1222003613  0.054253118 -0.2273329755 -0.1641308533 -0.359535991
##  [13,]  0.9350018518  0.460463529  0.2151945169 -0.5950749329  0.882275768
##  [14,]  0.2901525033  0.177233092 -0.0747665253  0.1572907547  0.491824198
##  [15,]  0.6126485250 -0.768490122 -0.8144434014 -0.2589326897 -0.320348448
##  [16,] -0.2643085665 -0.152614587  0.2645275795  0.1085583446 -0.542256181
##  [17,] -0.2977193090  0.108421055 -0.0511099657 -0.1517488314 -0.401022206
##  [18,] -0.2316107357  0.061675827 -0.1014745133 -0.3929930748 -0.106710066
##  [19,]  0.4390263003  0.903610115 -0.2511852038  0.8124390522  0.547488231
##  [20,] -0.0882574824 -0.367978294 -0.4070531166 -0.2884531971  0.204840995
##  [21,]  0.8113966403 -0.103615309 -0.0060245678 -0.9879546177 -0.452811107
##  [22,] -0.0681072013  0.001608724  0.0011192945 -0.3672914294 -0.141991848
##  [23,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
##  [24,]  0.0321922218 -0.053777539 -0.0867306264 -0.1753079280  0.228267904
##  [25,] -0.8188545986 -0.333879597  0.4495233879 -0.8452613227  0.866282480
##  [26,] -0.3357328520 -0.092149185  0.2368054809  0.7279500853 -0.286977959
##  [27,] -0.0595127801  0.149234753 -0.1440456177 -0.0041808611 -0.081003179
##  [28,]  0.1222003613  0.054253118 -0.2273329755 -0.1641308533 -0.359535991
##  [29,]  0.0750810722 -0.372355102  0.1724213631 -0.1125617938  0.365046270
##  [30,]  0.0344212939 -0.005581891  0.0573155874  0.5271596252 -0.351625136
##  [31,]  0.1883089347  0.007507890 -0.2776975231 -0.4053750968 -0.065223851
##  [32,] -0.0583380720  1.384326965 -0.2605871804 -0.0196630359  1.049868910
##  [33,]  0.2741033047 -0.186453595 -0.1607313795 -0.2085014685 -0.052508588
##  [34,] -0.1179740008  0.134488953  0.0890701236  0.3767084104 -0.471407917
##  [35,]  0.1222003613  0.054253118 -0.2273329755 -0.1641308533 -0.359535991
##  [36,] -0.7224401359  0.530383539 -0.0214890600 -0.0248325672 -0.333823699
##  [37,] -0.5012257476 -0.677949049 -0.1049064556  0.4367938980  0.180945607
##  [38,] -1.7613326934 -0.184052461 -0.7106529596  0.8770837823  1.038675761
##  [39,] -0.1468713872 -0.140027478 -0.1399867643 -1.0490734713 -0.126355653
##  [40,]  0.2314079172  0.297888425  0.3757782037  0.9017129116 -0.356906770
##  [41,] -0.3419236323 -0.442971227 -0.7358077795  0.7206943006  0.285280066
##  [42,]  0.9448956250  0.596105492 -0.0176306707  0.1703709334  0.532984599
##  [43,]  0.1563420437  0.712055073  0.2150232710  0.4833670538  0.101651718
##  [44,] -0.0801210804  0.073633163 -0.0369105978  0.3799358621  0.090151949
##  [45,]  0.2412392110 -0.602617353  0.2356279853  0.2667912006  0.634939518
##  [46,] -0.0339163516 -0.007032311 -0.0363660788  0.0659363154 -0.066044236
##  [47,] -0.1989001318  0.235235982  0.1064292461 -0.3960387477  0.190720633
##  [48,] -0.3278480705  0.159894115 -0.0594636391  0.1626141092  0.497972283
##  [49,]  0.3936744964 -0.003649504 -0.7611514191  0.5994389984  0.646827176
##  [50,]  0.6594663758  0.012153291  0.5248407035 -1.1410153634  0.184815286
##  [51,]  1.0775459707  0.264648268  0.2806500242  0.1960491627  0.016584101
##  [52,] -0.4460703250  0.261054673 -0.0157158892  0.0067082419  0.188958532
##  [53,]  0.2669030192  0.701579258 -0.2787070268  0.6544791729 -0.218745422
##  [54,] -0.0784196047  0.246155087  0.6457936294 -0.4624402246  1.088800384
##  [55,] -0.5240782374 -0.418446959 -0.3051267393 -0.2100953750  0.701148427
##  [56,] -0.1397631952 -0.268070389  0.2450871102 -0.4772051122 -0.113450578
##  [57,] -0.8452655260 -0.432722878  0.1130848563  0.1125103141  0.199584737
##  [58,]  0.3018946551 -0.449995638 -0.2218065985 -0.2128403547  0.459646642
##  [59,]  1.3859341267  0.045577295 -0.6749436895  0.1061040874 -0.106336399
##  [60,] -0.0533173297 -0.128699372 -0.1462712441 -0.2788626889  0.084169316
##  [61,] -0.1325736758  0.865811470  0.2099547422 -0.3751649276  0.699322533
##  [62,] -0.0240024286 -0.305494803 -0.5267882796 -1.0876874605  0.433883197
##  [63,] -0.0339163516 -0.007032311 -0.0363660788  0.0659363154 -0.066044236
##  [64,]  0.0004974229 -0.027729869  0.0860325617 -0.2232486015 -0.285699455
##  [65,] -0.2316107357  0.061675827 -0.1014745133 -0.3929930748 -0.106710066
##  [66,]  0.1553074823  0.155373425 -0.8817503451 -0.0853956854  0.369213843
##  [67,]  0.2541942554 -1.033247287  1.2794823779 -0.8761926382  0.991010304
##  [68,] -0.0434599167  0.097730213 -0.4378057665 -0.3155694358  0.048038066
##  [69,] -0.1091523733 -0.111853915 -0.0517185139 -0.1766819209  0.064560637
##  [70,] -0.2230480013 -0.807814536 -0.1836273933 -0.4872978663  0.506234629
##  [71,]  0.2162520271 -0.480013573 -0.1698196542 -0.3081914817  0.093958282
##  [72,]  0.2156229253 -0.541686723 -0.2003707100  0.4729180465 -0.368755533
##  [73,]  0.3621264146  0.498367591 -0.6848291791 -0.0446876958 -0.614184905
##  [74,]  0.1335319777  0.004127387 -0.5383907537 -0.2820508271 -0.023697394
##  [75,] -0.2738408331  0.178594738 -0.0994295954  0.0472847976 -0.064360783
##  [76,]  0.3286286260 -0.029469921 -0.0049944577 -0.0290420139 -0.005209685
##  [77,]  0.3225143990  0.264136532 -0.0671337782  0.2987975962  0.119348768
##  [78,]  0.3017945268  0.042999123 -0.1673649337 -0.1547801515 -0.580864230
##  [79,] -0.3410788755 -0.247845041  0.1422525932  0.4891372519 -0.122007490
##  [80,] -0.2119249390 -0.085540430  0.0658561779  0.0451247968 -0.388306943
##  [81,] -0.3635583526  0.015922232 -0.0149202717 -0.0446346444  0.277956138
##  [82,] -0.0336367297 -0.226491603 -0.1523951961 -0.1738599241 -0.169431069
##  [83,] -0.5973256921 -1.117492031 -0.4796315816  0.8349115323  0.283267534
##  [84,]  1.0507934833 -0.394880988  0.0153808584 -0.3419125817  0.036259220
##  [85,]  0.7843373696  0.483593599  0.7845136972 -0.6967361746 -0.373432459
##  [86,]  1.1401794776 -0.502249118  0.2216221377  1.3877099153 -0.368885224
##  [87,] -0.2316107357  0.061675827 -0.1014745133 -0.3929930748 -0.106710066
##  [88,]  0.3875888968 -0.150962362 -0.0503987902  0.0420934767 -0.568148967
##  [89,]  0.0321922218 -0.053777539 -0.0867306264 -0.1753079280  0.228267904
##  [90,]  0.2428769311 -0.218324478  0.2046931499  0.3983197357 -0.140277714
##  [91,] -0.2316107357  0.061675827 -0.1014745133 -0.3929930748 -0.106710066
##  [92,] -0.0339163516 -0.007032311 -0.0363660788  0.0659363154 -0.066044236
##  [93,]  0.1883089347  0.007507890 -0.2776975231 -0.4053750968 -0.065223851
##  [94,]  0.0321922218 -0.053777539 -0.0867306264 -0.1753079280  0.228267904
##  [95,]  0.3666317483 -0.243942489  0.0060935061  0.0267457255 -0.061294039
##  [96,]  0.0065957932  0.102489525 -0.1944101653 -0.2454251045  0.213308960
##  [97,]  0.1879182315 -0.256905869 -0.0283911690 -0.1099529169  1.574624107
##  [98,]  0.5100613548 -0.941913933 -0.5189138892 -0.8630731209 -0.804542418
##  [99,]  0.8339530114  0.978253414  0.2407886742 -0.2969091760  0.589427004
## [100,]  0.4212646049  1.046791046  0.1829324473 -0.8275178582 -0.243163690
## [101,] -0.3480302807  0.116293652  0.0163289420 -0.2237278213 -0.139598317
## [102,] -0.5551927751 -0.708417635  0.4516179803  0.5061712067  0.138568241
## [103,]  0.4239259555  0.490703655  1.2408523773 -0.4005348892  0.137862931
## [104,]  0.5632664802  0.349170570 -0.2488482900 -0.0972637652  0.091282795
## [105,] -0.0967207082  0.334679541 -0.0736222912  0.1236201280 -0.375194639
## [106,] -0.3761236783 -0.396003658 -0.2556969022  0.4043604047  0.266302975
## [107,] -0.2058003030  0.104169811  0.0592587480  0.0776815113 -0.003225376
## [108,] -0.0887378109 -1.181188883 -0.2793338637 -0.0796797588  0.085106801
## [109,]  0.0030302426 -0.136042779  0.0848640890 -0.0135849706  0.749905052
## [110,]  0.0077613932 -0.777842494  0.7657597610 -0.0718781073  0.658082683
## [111,]  0.0099632115  0.032214368 -0.1090599426 -0.3711703214 -0.128674342
## [112,] -0.0355858979  1.519147737  0.6745755287 -0.1379321299  0.407385718
## [113,]  0.5084418901  0.334011167  0.0727546935  0.2463280687  0.149538748
## [114,]  0.1899240629  0.259322835  0.2890315581 -0.1093807422  0.492114094
## [115,]  0.0835136934 -0.522188059 -0.2124265832 -1.0245554469  0.271778678
## [116,]  0.0805386413 -0.317042897  0.6407119859 -1.0188811688 -0.616020739
## [117,]  0.3179776146  0.042067017  0.1425460938  0.3063206205  0.721700563
## [118,] -0.3250010065 -0.175108454 -0.0095131510 -0.1778096307  0.070653203
## [119,]  0.0015670468 -0.193121199  0.1480389725  0.1908309537  0.208094916
## [120,] -0.6211658596 -0.146558527 -0.0318897850 -0.5370016190  0.108150663
## [121,] -0.1563491662  0.710201454  0.0907603626  0.2525830607  0.346166022
## [122,]  0.1178754336  0.122773150 -0.3444797068  0.2611483527  0.450376156
## [123,]  0.7337254752 -0.222150659 -1.5790648285  0.3251285347  0.730642874
## [124,] -0.0339163516 -0.007032311 -0.0363660788  0.0659363154 -0.066044236
## [125,]  0.2084093388 -0.375100100 -1.1038508919 -0.0726128805  0.291372494
## [126,]  0.0645541175  0.033125900 -0.0790978792 -0.0338010865 -0.144207526
## [127,]  0.4004563675 -0.458766082  0.9905808141  0.1380440645  0.244537805
## [128,]  0.2863332879  0.180722237  0.2296462157 -0.1037233757  0.476306562
## [129,] -0.1091523733 -0.111853915 -0.0517185139 -0.1766819209  0.064560637
## [130,]  0.1883089347  0.007507890 -0.2776975231 -0.4053750968 -0.065223851
## [131,] -0.3336358473 -0.211092552 -0.6757519936  0.0703996050  0.423826616
## [132,] -0.0772065943  0.231378880 -0.3798084718  0.0671816492 -0.618749504
## [133,]  0.1780397096  0.068617133  0.0312347102  0.2167938587 -0.659847905
## [134,] -0.1846891363  0.092024587 -0.2086905038 -0.5784789128  0.426341372
## [135,] -0.2977193090  0.108421055 -0.0511099657 -0.1517488314 -0.401022206
## [136,]  0.1570825611 -0.024362994  0.0877270064  0.2014461075 -0.152992977
## [137,] -0.4141388540  0.163038880  0.0666934896  0.0175164222 -0.433910456
## [138,]  0.1570825611 -0.024362994  0.0877270064  0.2014461075 -0.152992977
## [139,]  0.2808373783 -0.049981004 -0.1108726375 -0.1701279028 -0.074009301
## [140,]  0.6395335978  0.439930517 -0.0917460072 -0.2851020209  0.624067273
## [141,]  0.1899240629  0.259322835  0.2890315581 -0.1093807422  0.492114094
## [142,]  0.1222003613  0.054253118 -0.2273329755 -0.1641308533 -0.359535991
## [143,] -0.1682454354  0.018445611  0.0207164246  0.6641711027 -0.852496280
## [144,]  0.1262768358 -0.139953366 -0.0863032023 -0.2695119870 -0.137158923
## [145,]  0.0097890850 -0.009974093 -0.1111662015  0.7578026459  0.521713143
## [146,] -0.1669298068 -0.497676355  0.5385750613  0.3094021887  0.453349689
## [147,]  0.0681679715 -0.139230559 -0.0006817074  0.1444085403  0.315162433
## [148,] -0.8045907292  0.292440645 -0.2815616993 -0.1753618123 -0.066690728
## [149,]  0.6471594658  1.085340725  0.3762444312  0.2976880208  0.410435663
## [150,]  0.1188779895  0.235197565  0.0773144734  0.1748528019  0.148060778
## [151,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [152,]  0.7927847745  0.218889530 -0.1853557986  0.1256441311 -0.209472605
## [153,] -0.5204900424 -0.038981780  0.2609272701  0.4359920290 -0.034285309
## [154,]  0.1883089347  0.007507890 -0.2776975231 -0.4053750968 -0.065223851
## [155,]  0.6581642295 -0.270700206  1.3359808385  0.7099965412 -0.521547233
## [156,] -0.5372910768  0.829746503  0.4269331574  0.4286759846  0.278541817
## [157,] -0.3072628741  0.213183579 -0.4525496534 -0.5332545825 -0.286939904
## [158,]  0.0645541175  0.033125900 -0.0790978792 -0.0338010865 -0.144207526
## [159,]  0.4850883253 -0.493554822  0.2471774255  1.2510482703  0.757597234
## [160,] -0.3266638522  0.091516518 -0.5624548187 -0.8780535868 -0.136726352
## [161,] -0.2925633117  0.237845088 -1.1232074733  0.5209197267  0.588077639
## [162,] -0.1378721038 -0.259797103  0.1675594084 -0.6275321863 -0.453682799
## [163,]  0.0645541175  0.033125900 -0.0790978792 -0.0338010865 -0.144207526
## [164,]  0.2464907851 -0.390680937  0.2831093218  0.5765293625  0.319092246
## [165,]  0.1570825611 -0.024362994  0.0877270064  0.2014461075 -0.152992977
## [166,]  0.0321922218 -0.053777539 -0.0867306264 -0.1753079280  0.228267904
## [167,] -0.0339163516 -0.007032311 -0.0363660788  0.0659363154 -0.066044236
## [168,] -0.3573830425 -0.878386106 -0.2486403987  0.2775893340  0.096622041
## [169,]  0.6095286349 -0.853745103  0.8162144210  0.5321258565 -0.463139279
## [170,] -0.4841610561  0.527396728 -0.4448977941 -0.0354413482  0.473134202
## [171,]  0.1883089347  0.007507890 -0.2776975231 -0.4053750968 -0.065223851
## [172,] -0.0756578464  0.471156474 -0.6003872604  0.5796401393  0.816987862
## [173,] -0.2316107357  0.061675827 -0.1014745133 -0.3929930748 -0.106710066
## [174,]  0.0397599866 -0.183577518  0.1162153258 -0.5186989717  0.418831432
## [175,]  0.0681679715 -0.139230559 -0.0006817074  0.1444085403  0.315162433
## [176,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [177,] -0.0882574824 -0.367978294 -0.4070531166 -0.2884531971  0.204840995
## [178,] -0.0261734111 -0.469112391  0.0913383574  0.1468043480 -0.290625602
## [179,] -0.1336785617 -0.450919163  0.6056862726 -0.0002510536  0.553366214
## [180,]  0.2242846844 -0.077945130 -0.1916486041 -0.0856586284  0.021670678
## [181,]  0.5308355006  0.234290069  0.1402984183  0.2843960742 -0.178226614
## [182,] -0.5284490608 -0.790091003  0.1363720099 -0.4879766299 -0.819638473
## [183,] -0.1049938663  0.719996355  0.2127875264 -0.2421671820 -0.520803429
## [184,] -0.3876015173 -0.401433501  0.1850863710 -0.1470185277  0.438478898
## [185,] -0.8159759481  0.462966077  0.0393374909  0.1496128763 -0.609894574
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## [567,]  0.7641098758  0.770244791  0.0301422631  0.3963239828 -0.084515281
## [568,]  0.2063133333 -1.144034669 -0.4861682616  0.2633728622 -0.144071414
## [569,]  0.1570825611 -0.024362994  0.0877270064  0.2014461075 -0.152992977
## [570,]  0.6879584111  0.384627769  0.0537702971  0.4094680747  0.020782941
## [571,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [572,] -0.5300348353 -0.606610969  0.6307670164 -0.2364383452 -0.642108970
## [573,] -0.1351649185  0.300742505 -0.4951207578 -0.1444423689 -0.261233017
## [574,] -0.0923657719 -0.407502184 -0.9121532368  1.1196574078  0.776439982
## [575,]  0.0991242369  0.045000632 -0.0275852797 -0.0101779105  0.204523510
## [576,] -0.5514588734 -0.376392435  0.0265524187 -0.7537430200 -0.313885280
## [577,] -0.1517485597 -0.003148127 -0.2512751139 -0.0666228784  0.301452599
## [578,]  0.5234954223  0.222344997 -0.4931514094 -0.2907169058 -0.557790328
## [579,] -0.0172953081  0.099618457  0.0902181756  0.1590873430  0.171635259
## [580,] -0.3415043285 -0.315890102  0.3329334598 -0.3390113465  0.056002220
## [581,] -0.0595127801  0.149234753 -0.1440456177 -0.0041808611 -0.081003179
## [582,] -0.4141388540  0.163038880  0.0666934896  0.0175164222 -0.433910456
## [583,] -0.0430437999 -0.158599143 -0.1020830615 -0.4179261643  0.358872777
## [584,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [585,] -0.4749786407  0.058607369  0.1748900163  0.4916775001  0.357748669
## [586,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [587,]  0.0020593982 -0.092485331  0.0496828402  0.3856527837  0.020850294
## [588,] -0.2977193090  0.108421055 -0.0511099657 -0.1517488314 -0.401022206
## [589,] -0.4558730926  0.083503571  0.0530413947  0.2229944537  0.122482939
## [590,]  0.1196636853  0.503923329  0.3310838233  0.3755103414  0.058996765
## [591,]  0.0696249296  0.578366891  0.1536582758  0.1233759696 -0.045722636
## [592,] -0.0532879220 -0.189774551  0.1823165159  0.0391277474 -0.102780253
## [593,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [594,] -0.1890681159  0.284708436  0.5467533798 -0.6509586818 -0.013869645
## [595,]  0.0991242369  0.045000632 -0.0275852797 -0.0101779105  0.204523510
## [596,] -0.3888933352 -0.126629876  0.2853213839  0.2652047125 -0.452970163
## [597,]  0.3033215699  0.643562001  0.2512571205 -0.5654743538 -0.182296612
## [598,] -0.3598003511 -0.977780393  0.6425107545 -0.5384148925 -0.536254009
## [599,]  0.4245473974  0.069509928 -0.2052479591 -0.3858656344 -0.296457759
## [600,]  0.1581761111 -0.031199902 -0.1412840565  0.1555856150 -0.272641461
## [601,] -0.3473589022 -0.152667614  0.0412171381 -0.3242498912 -0.255458389
## [602,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [603,] -0.2977193090  0.108421055 -0.0511099657 -0.1517488314 -0.401022206
## [604,] -0.0595127801  0.149234753 -0.1440456177 -0.0041808611 -0.081003179
## [605,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [606,]  0.4679295571 -0.983408093  0.1530848915 -0.4669342353  0.324220735
## [607,] -0.0293902906  0.296463014  0.1579576249  0.1005617765 -0.043181400
## [608,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [609,]  0.0622573396 -0.130179751  0.0589694448  0.5124092354  0.018606613
## [610,]  0.5771078256 -0.456365729  0.3237695797  0.1245442090 -0.557277556
## [611,] -0.7816939936 -0.349708701 -1.7039745292  0.1514414268  0.121346293
## [612,]  0.3460407758  0.320608264  0.0980646613 -0.3394479109  0.198622339
## [613,]  0.2808373783 -0.049981004 -0.1108726375 -0.1701279028 -0.074009301
## [614,] -0.1390822921  0.004186933  0.0653503723 -0.1577458808 -0.115495516
## [615,] -0.4155227778 -0.773123149  0.0315887664 -0.0249166028  0.406266539
## [616,] -0.0595127801  0.149234753 -0.1440456177 -0.0041808611 -0.081003179
## [617,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [618,]  0.2705736627  0.313712282  0.5800014669  0.4748598875  0.549976193
## [619,]  0.1570825611 -0.024362994  0.0877270064  0.2014461075 -0.152992977
## [620,] -0.0828419204  0.161264056  0.0246634898  0.1827945897 -0.151309524
## [621,] -0.6561854980  0.843996879 -0.8405559233  0.0125510353 -0.391507926
## [622,] -0.8944883105  0.259373905  0.1633150054 -0.3174545662  0.112952675
## [623,] -0.1390822921  0.004186933  0.0653503723 -0.1577458808 -0.115495516
## [624,]  0.1318348408  0.218560787  0.1803184797 -0.0132235834  0.501954209
## [625,] -0.1063716881  0.177747088  0.2732541317 -0.1607915537  0.181935183
## [626,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [627,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [628,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [629,]  0.2808373783 -0.049981004 -0.1108726375 -0.1701279028 -0.074009301
## [630,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [631,] -0.3618530254 -0.143215902  0.3773876081  0.4299949796  0.251241380
## [632,]  0.1574312693  0.062293722  0.2879980186  0.0568935931  0.516913152
## [633,]  0.1154015729 -0.184977026  0.3652961670 -1.2710501954  0.291489970
## [634,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [635,]  0.7125417888 -0.389238545  0.3484086194  0.4939188970 -0.690126110
## [636,] -0.3713598935  0.079960550  0.3510275742  0.2636811026  0.296033641
## [637,] -0.1351649185  0.300742505 -0.4951207578 -0.1444423689 -0.261233017
## [638,] -0.2977193090  0.108421055 -0.0511099657 -0.1517488314 -0.401022206
## [639,]  0.0645541175  0.033125900 -0.0790978792 -0.0338010865 -0.144207526
## [640,] -0.0015544559  0.079871128 -0.0287333316  0.2074431569 -0.438519666
## [641,] -0.0595127801  0.149234753 -0.1440456177 -0.0041808611 -0.081003179
## [642,] -0.4863317331 -0.118440340 -0.3617832382  0.3497605752  0.395287144
## [643,]  0.4476769581 -0.145809216 -1.2102375812  0.0072859136 -0.708708162
## [644,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [645,] -0.0339163516 -0.007032311 -0.0363660788  0.0659363154 -0.066044236
## [646,] -0.2316107357  0.061675827 -0.1014745133 -0.3929930748 -0.106710066
## [647,] -0.0835559447 -0.268120980  0.0559610250 -0.1065647444  0.079519581
## [648,] -0.0835559447 -0.268120980  0.0559610250 -0.1065647444  0.079519581
## [649,]  0.0311561481  0.253431283  0.1791704278  0.2043974840 -0.141088967
## [650,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [651,] -0.3478598128 -0.100244847  0.3861296937  0.8707509780  0.405193884
## [652,]  0.0645541175  0.033125900 -0.0790978792 -0.0338010865 -0.144207526
## [653,] -0.2406536338  0.209007082 -0.7778506356  0.3004603079  0.029143083
## [654,] -0.0636841178  0.429038822  0.3037467627 -0.0421598625  0.103037642
## [655,] -0.6478026910  0.784745065  0.3655604860 -0.2972143214 -0.315619370
## [656,] -0.2198056584 -0.247402878  0.2173581579  0.3638630018 -0.066845086
## [657,]  0.1883089347  0.007507890 -0.2776975231 -0.4053750968 -0.065223851
## [658,]  0.4646458934  0.103621273  0.1722712338 -0.0532967362  0.013385252
## [659,] -0.0339163516 -0.007032311 -0.0363660788  0.0659363154 -0.066044236
## [660,] -0.3790067736  0.189813983  0.0022868558 -0.1763973986 -0.113812064
## [661,] -0.1503358965  0.047585513  0.0814373765  0.2352015689 -0.098932487
## [662,]  0.0991242369  0.045000632 -0.0275852797 -0.0101779105  0.204523510
## [663,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [664,]  0.2808373783 -0.049981004 -0.1108726375 -0.1701279028 -0.074009301
## [665,] -0.0307630506 -0.166582934  0.6846350399  0.0061394152 -0.465961294
## [666,] -0.6285125252 -0.033921920 -0.2091292468 -0.1783660670  0.577856026
## [667,] -0.0690563452  0.253997277 -0.5454853054 -0.3856866123  0.033079122
## [668,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [669,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [670,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [671,]  0.1247206654 -0.111266433  0.0800942592  0.0599392660  0.219482453
## [672,] -0.1734384073  0.191073250  0.0668914952  0.2191883529 -0.375700806
## [673,] -0.1390822921  0.004186933  0.0653503723 -0.1577458808 -0.115495516
## [674,]  0.2770363816 -0.314902041  0.8160033180 -0.3230940622  0.259014835
## [675,]  0.1574312693  0.062293722  0.2879980186  0.0568935931  0.516913152
## [676,]  0.2826069530  0.516806166 -0.0970999746 -0.8574676374  0.516253557
## [677,]  0.1570825611 -0.024362994  0.0877270064  0.2014461075 -0.152992977
## [678,] -0.0477591847  0.160536602 -0.0292778506  0.5214427036 -0.282323481
## [679,]  0.3172757484 -0.157284049  0.0377898829 -0.1340021949 -0.224107664
## [680,]  0.2808373783 -0.049981004 -0.1108726375 -0.1701279028 -0.074009301
## [681,]  0.4528945748  1.169861743 -0.3788482615  0.8891812797  0.734126710
## [682,] -0.2501528114 -0.553383229  1.0039160238 -0.7550992193 -0.886971413
## [683,] -0.3689824494  0.139614158  0.4021793636 -1.5257446772 -0.079359252
##               [,24]         [,25]        [,26]        [,27]        [,28]
##   [1,] -0.023870906 -0.2533762299  0.060457998  0.097935932 -0.030603065
##   [2,] -0.734807029  0.0536516959 -0.133515096  1.282100078 -0.396582018
##   [3,] -0.042387890 -0.7096277444 -0.006793177 -0.281299221 -0.465836655
##   [4,]  0.522366075 -0.2001565386 -0.335246573  0.127504221 -0.276401059
##   [5,] -0.259186950 -0.3030214681 -0.047545892 -0.053492687  0.207683314
##   [6,]  0.517170481  0.4928020473  0.969483524  1.372102760 -1.149202990
##   [7,]  0.036512316 -0.1295981594 -0.102299087  0.220646090 -0.209412323
##   [8,] -0.110976507  0.1216312254  0.289399359 -0.147099555  0.100490940
##   [9,] -0.489072424  0.9632167877  0.920840864 -0.418364621  0.158732337
##  [10,] -0.104260180  0.1098541111 -0.307757594 -0.073090989  0.402720550
##  [11,]  0.238664209 -0.0341451650 -0.456332573  0.194267031 -0.420041224
##  [12,] -0.114007166  0.1532870821  0.395142814 -0.259196287  0.080682723
##  [13,]  0.319201024  0.1958002999  0.208092648 -0.129411639 -0.163797518
##  [14,]  0.158357289 -0.2189931319 -0.408828945  0.089887101 -0.566215726
##  [15,]  0.061009314 -0.3176369973 -0.511175376  0.107447198 -0.312449234
##  [16,] -0.098598534  0.0750372533  0.431666352  0.142971252  0.270744129
##  [17,] -0.117550435 -0.0007865872  0.040374971 -0.198187052  0.222639665
##  [18,] -0.112717010 -0.1014343356 -0.059514155 -0.044993530  0.127094537
##  [19,] -0.150613784 -0.2751729659 -0.064137046 -0.571733476  0.031836876
##  [20,]  0.090631379 -0.1451354783 -0.016376134 -0.175802417  0.162670813
##  [21,]  0.118125181  0.8189205963 -0.295000240 -0.172516804 -0.416562238
##  [22,]  0.319905194 -0.4429985171 -1.151641353  0.170536275  0.246439297
##  [23,] -0.091279787 -0.0823431037  0.073087176 -0.403882225 -0.002428060
##  [24,]  0.047040059 -0.1394898905 -0.138522724  0.185136766 -0.152831092
##  [25,] -0.579381021 -0.1770551337 -1.140255927  0.626232544  0.332025567
##  [26,]  0.113634617  0.1336423200 -0.592855239 -0.269655898 -0.089557308
##  [27,] -0.028704331 -0.1527284815  0.160347123 -0.055257590  0.064942064
##  [28,] -0.114007166  0.1532870821  0.395142814 -0.259196287  0.080682723
##  [29,]  0.324169704  0.1486729796  0.021044669  0.511455342  0.150006774
##  [30,]  0.100344363  0.0230016217 -0.244722674 -0.394751960 -0.269638193
##  [31,] -0.109173740  0.0526393337  0.295253688 -0.106002765 -0.014862406
##  [32,]  0.049121687  0.2022624292  0.059662061 -1.312948675  0.801642457
##  [33,] -0.101858049  0.0595299291  0.436274235 -0.016241110 -0.205260857
##  [34,] -0.093082554 -0.0133512120  0.067232846 -0.444979015  0.112925285
##  [35,] -0.114007166  0.1532870821  0.395142814 -0.259196287  0.080682723
##  [36,] -0.183693865  0.0537629239  0.299432862  0.124018809  0.111671962
##  [37,]  0.313540495  0.2380725450 -0.465755112  0.040058029  0.215544592
##  [38,] -0.686327980  0.1834504251  0.615906495  0.133231800 -0.171629830
##  [39,]  0.644875721 -0.3767509584 -0.368889163  0.194683899  0.617559651
##  [40,]  1.178878565 -0.3516417461 -0.180390275 -0.187185854 -0.236351204
##  [41,] -0.277001701  0.1170510356  0.447545846  0.017653922  0.803604430
##  [42,]  0.290613335 -0.8608805199 -0.496353632  0.133806429  0.126061313
##  [43,] -0.771941237 -0.2216764549 -0.048501273 -0.055363776  0.550505138
##  [44,]  0.160992503  0.2025385482  0.076433464 -0.079869995 -0.171077897
##  [45,] -0.629844999 -0.6261891527  0.524982949 -1.054463203  0.810491383
##  [46,]  0.042206634 -0.0388421421 -0.038633598  0.031943244 -0.057285963
##  [47,] -0.259186950 -0.3030214681 -0.047545892 -0.053492687  0.207683314
##  [48,]  0.184738026  0.0797408995  0.422336122  0.405159893 -0.209387867
##  [49,] -0.461696409  0.5897743894 -0.324710381 -0.939213721 -0.189689964
##  [50,]  0.394408604  0.3205115569  0.060643007 -0.267774701  0.333508974
##  [51,]  0.370307861  0.7098679030  1.020747056 -0.395090672 -0.345067391
##  [52,]  0.984293844  0.6127453325  0.192267229  0.744178251 -1.147314716
##  [53,]  0.211112074 -0.3224540600 -0.617954091  0.134493392  0.177130333
##  [54,] -0.561559361 -0.5800157139  0.457624867 -0.529682759  0.025369586
##  [55,] -0.926063148  1.6862959842 -0.501492946 -0.717024868  0.083918159
##  [56,]  0.193556218  0.4741982033 -0.839036558 -0.095580736  0.034354113
##  [57,]  0.684181977 -0.1361702848 -0.174370790  0.278950592  0.213566298
##  [58,]  0.153416377 -0.4951128672 -0.643573472 -0.334412354 -0.517039603
##  [59,]  0.196738880  0.4538104022 -0.574758894  0.217582549 -0.272960396
##  [60,]  0.086265105 -0.5654790344 -0.018624826  0.001332726 -0.425149429
##  [61,] -0.149699225  0.4578781242 -0.422110942  0.300507172  0.214099122
##  [62,] -0.146429890 -0.0716086514 -0.052296040 -0.400198958 -0.125074704
##  [63,]  0.042206634 -0.0388421421 -0.038633598  0.031943244 -0.057285963
##  [64,]  0.394465201 -0.5629492746  0.167038804 -0.055292217 -0.120566575
##  [65,] -0.112717010 -0.1014343356 -0.059514155 -0.044993530  0.127094537
##  [66,] -0.531128358  0.1260123457  0.259157778 -0.287604053  0.454956264
##  [67,] -1.739396357 -0.0257798199  0.281181813 -0.590066405  0.032267556
##  [68,] -0.355858250 -0.3285573970 -0.143575716  0.368585727 -0.421346342
##  [69,]  0.210899710 -0.1363468170  0.277238178  0.175762703  0.170207428
##  [70,]  1.065158349 -0.9732618741  0.797973162  0.134276060  0.572775189
##  [71,] -0.028544578  0.0620813181  0.571002534  0.786150175 -0.424468447
##  [72,] -0.043349164 -0.5026445505 -0.180531753  0.525399121  0.881410649
##  [73,] -0.055750661 -0.4959490020 -0.541568869 -0.058840913  0.499926741
##  [74,] -0.401574729 -0.6322512897  0.006955194  0.395299678 -0.437732386
##  [75,]  0.154485338 -0.1012590602 -0.061831964 -0.005213843  0.126556024
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## [657,] -0.109173740  0.0526393337  0.295253688 -0.106002765 -0.014862406
## [658,]  0.135852393  0.4719342408  0.259802565  0.066973485  0.022346905
## [659,]  0.042206634 -0.0388421421 -0.038633598  0.031943244 -0.057285963
## [660,]  0.037087297  0.1079299518 -0.040036182  0.013147665  0.508509024
## [661,]  0.040403867  0.0301497496 -0.044487928 -0.009153546  0.058067382
## [662,]  0.013654698  0.0184049756  0.103134336  0.193234213  0.166969436
## [663,]  0.084565663  0.1322913150 -0.095846385  0.280435048  0.044741409
## [664,] -0.071648137  0.3244205392  0.337930027 -0.010704483  0.182710095
## [665,] -0.124836421  0.6947689941 -0.236143201  0.017505659 -1.146550819
## [666,] -0.764468235 -0.0664689635 -0.162458995 -0.093336327  0.123383594
## [667,] -0.426769215 -0.4424437364  0.055405006  0.281384893 -0.299118315
## [668,]  0.084565663  0.1322913150 -0.095846385  0.280435048  0.044741409
## [669,]  0.084565663  0.1322913150 -0.095846385  0.280435048  0.044741409
## [670,]  0.084565663  0.1322913150 -0.095846385  0.280435048  0.044741409
## [671,]  0.084565663  0.1322913150 -0.095846385  0.280435048  0.044741409
## [672,] -0.106307035  0.0019905265  0.209191895 -0.289224004 -0.026624169
## [673,] -0.075191407  0.1703468699 -0.016837816  0.050304752  0.324667037
## [674,] -0.498380859  0.3535439711 -0.099871558  0.603651367 -0.042836580
## [675,] -0.061904278 -0.0692958175 -0.083878122  0.271935891  0.125330186
## [676,] -0.168457816  0.8423920070  0.440815993 -0.463548099  0.485333163
## [677,] -0.048920758  0.0887903534  0.015874389 -0.155390421  0.099599312
## [678,]  0.027506083  0.1590375866  0.188154237 -0.515695464 -0.116219994
## [679,]  0.056738949 -0.2534530209 -0.041697493 -0.286313257 -0.470247985
## [680,] -0.071648137  0.3244205392  0.337930027 -0.010704483  0.182710095
## [681,] -0.634813932  0.1193984778  0.024809098  0.586658623 -0.705833228
## [682,] -0.476955375  0.2110308048 -0.753006810 -0.325166516 -0.018335915
## [683,]  0.263563896  0.8078837253 -0.459568662 -0.748049933  0.175047539
##                [,29]        [,30]         [,31]         [,32]
##   [1,] -0.1053899525 -0.221954475  0.0374261817 -0.0898273461
##   [2,] -0.0751020225 -0.481565372  0.0776935703  0.2145103167
##   [3,] -0.4015402900  0.203911264 -0.0665100485  0.1220074534
##   [4,] -0.4173451286 -0.356796758 -0.3007207052 -0.0851070251
##   [5,]  0.0032400817  0.435227366  0.1390762049 -0.2047073743
##   [6,] -0.4948010198  0.272584182 -0.2738849352 -0.1873908052
##   [7,] -0.1241490511 -0.232431729  0.0414691624 -0.0979506549
##   [8,] -0.2512200926 -0.073287090 -0.3546910035 -0.4399262075
##   [9,] -0.3810239802  0.204952836 -1.0275737592  0.2356592827
##  [10,]  0.2818108005  0.063482097  0.1132852500 -0.0515562152
##  [11,]  0.1325902764 -0.240058029  0.2995559584  0.2242427719
##  [12,] -0.0257522848  0.214482653 -0.0754882177 -0.1240230785
##  [13,] -0.1126024468  0.428034555  0.7547233699  0.4142078103
##  [14,] -0.2661359017 -0.173009527  0.1417812491 -0.1207641047
##  [15,] -0.7928804790 -0.290051504 -0.2814408549  0.1074637876
##  [16,]  0.2474310824 -0.271810955 -0.5413026395  0.1470233583
##  [17,]  0.2236790945  0.099295026  0.2797190025 -0.1014961284
##  [18,] -0.0761610172 -0.070575249  0.1577961833 -0.2504835052
##  [19,] -0.4642169371 -0.143970584 -0.7660059361 -0.0442084364
##  [20,] -0.0296565433 -0.227837832 -0.0465038657 -0.0705726322
##  [21,]  0.1016006543 -0.293457879  0.1698818714  0.0038952512
##  [22,]  0.1762758808  0.863450486  0.0098311502 -1.0475972817
##  [23,] -0.1784922599  0.031725017  0.2664498449  0.1311074192
##  [24,] -0.2032290037 -0.283987564  0.0588755089 -0.0676506530
##  [25,] -0.6906673019  0.385430162  0.2305488002 -0.3960389618
##  [26,]  0.2154587262  0.039841623  0.3406965418  0.4729407571
##  [27,]  0.1944501592 -0.052084200  0.1593490009  0.0591600307
##  [28,] -0.0257522848  0.214482653 -0.0754882177 -0.1240230785
##  [29,] -0.0974817587 -0.095339782 -0.0194266685  0.1292605073
##  [30,]  0.1573270202  0.075654552  0.5071302944  0.4230008440
##  [31,] -0.3255923965  0.044612379 -0.1974110369 -0.2730104553
##  [32,] -0.4834626735  0.846177941  0.5608406082  0.2519352524
##  [33,] -0.3802390062  0.341908619 -0.0573904050 -0.4395438526
##  [34,] -0.1041199560 -0.086174451  0.1091698782 -0.0358083330
##  [35,] -0.0257522848  0.214482653 -0.0754882177 -0.1240230785
##  [36,]  0.6049396061 -0.090167016  0.2792201685 -0.0075614769
##  [37,]  0.5751560106 -0.076506729 -0.0660743156  0.5015721483
##  [38,] -0.4809422430  0.520516599  0.1302068429 -0.0950991549
##  [39,] -0.1833897306  1.072581436 -0.1453477817 -0.4094177796
##  [40,] -0.6009060171  0.624200484  0.2089526956  0.3564720546
##  [41,]  0.3152254003  0.292917192  0.4695441953  0.3692958948
##  [42,] -0.1294717549 -1.029307414  0.0022812476 -0.2599987379
##  [43,]  0.5012970972  0.219916197  0.2667543852 -0.1459489925
##  [44,] -0.0893040009 -0.304303641  0.1926267236 -0.4025231300
##  [45,] -1.0232313106  0.238451388 -0.5513826836  0.2438233275
##  [46,]  0.0966111080 -0.114117289  0.1807983282  0.0813367239
##  [47,]  0.0032400817  0.435227366  0.1390762049 -0.2047073743
##  [48,] -1.1549955208  0.129419943  0.2398569930 -0.6896193440
##  [49,] -0.0357231312  0.189203174  0.0364334460 -0.0074013507
##  [50,] -0.5986420221 -0.664192515  0.2386325673  0.4107544016
##  [51,]  0.1127907685  0.104835525  0.2164657449 -0.4780841419
##  [52,]  0.6667563271 -0.584299288 -0.3386665540 -0.2907265182
##  [53,]  0.3213715584 -0.353887385  0.0228768014  0.3883782224
##  [54,] -0.4385152552 -0.108477323  0.4151792174  0.0910601457
##  [55,]  0.1924214505 -0.368085298  0.4580880556  0.8364887561
##  [56,] -0.1605341894 -0.448324997  0.0226999155  0.6077664693
##  [57,]  0.0178296252  0.027523714  0.1103858064  0.0420250489
##  [58,]  0.1538286902  0.479380519  0.0539281602 -0.9219400334
##  [59,]  0.5053951762 -0.143237196  0.2709673205 -0.3910637709
##  [60,]  0.1734031896  0.227939232 -0.0302387460  0.2212241349
##  [61,] -0.8415261796 -0.241174867  0.1813684780  0.4161540751
##  [62,]  1.1926485979  0.125679122 -0.2878582073  0.3984561712
##  [63,]  0.0966111080 -0.114117289  0.1807983282  0.0813367239
##  [64,] -0.1421759966  0.307529819 -0.5546802497  0.3137682707
##  [65,] -0.0761610172 -0.070575249  0.1577961833 -0.2504835052
##  [66,]  1.4591949914  0.075714692  0.0894841985  0.0367071954
##  [67,]  0.9563527637 -0.288035215  0.5620963782  0.4344602919
##  [68,] -0.0991908256 -0.378315992  0.0425407636  0.1236996265
##  [69,]  0.1123182610 -0.063808707  0.4299455635  0.2305618715
##  [70,] -0.2952921779  0.164304658 -0.4481917745  0.2759311920
##  [71,]  0.1534926167  0.102686777  0.4787126567  0.3753808399
##  [72,] -0.3805169635 -0.119350115  0.4190183081 -1.1096112878
##  [73,] -0.6449382713 -0.643953759 -0.1882547526  0.1233945633
##  [74,]  0.2395482500 -0.007199791 -0.4633834721  0.1943933752
##  [75,]  0.3261311670 -0.216311380  0.0210182034  0.0677011772
##  [76,]  0.3184039319  0.297428578  0.4187414782 -0.0908720391
##  [77,] -0.5412392696 -0.027167220  0.2274327658 -0.0709934094
##  [78,] -0.0337186239  0.318331469 -0.0835673336 -0.0140203707
##  [79,]  0.3535924614  0.078044014 -0.3742614534  0.1322185730
##  [80,]  0.1690324848  0.396591266  0.4197396344 -0.2680295257
##  [81,] -0.0472582464 -0.257767404 -0.1290075709 -0.0398874329
##  [82,] -0.1906787222  0.347125762 -0.1603271952  0.3523188959
##  [83,]  0.2989536110 -0.211471606 -0.3339584280 -0.8367073610
##  [84,] -0.4230358180 -0.489606914 -0.9599246182  0.5702175843
##  [85,] -0.4063607095  0.191832489  0.2477765169  1.1515154163
##  [86,] -0.6081388117  0.393624450  0.1624156372  1.2515517165
##  [87,] -0.0761610172 -0.070575249  0.1577961833 -0.2504835052
##  [88,] -0.0883652337  0.615627710  0.0564532982 -0.1805537679
##  [89,] -0.2032290037 -0.283987564  0.0588755089 -0.0676506530
##  [90,] -0.3450998381  0.359523273 -0.0643510825 -0.1589040354
##  [91,] -0.0761610172 -0.070575249  0.1577961833 -0.2504835052
##  [92,]  0.0966111080 -0.114117289  0.1807983282  0.0813367239
##  [93,] -0.3255923965  0.044612379 -0.1974110369 -0.2730104553
##  [94,] -0.2032290037 -0.283987564  0.0588755089 -0.0676506530
##  [95,] -0.1923598630  0.542280909 -0.4062891451 -0.4140345330
##  [96,] -0.1053899525 -0.221954475  0.0374261817 -0.0898273461
##  [97,]  0.4908176077  0.090031961  1.0543621323  0.2140108562
##  [98,]  1.2350187765 -0.147587562 -0.6478564444  0.5047740558
##  [99,] -0.0250470935 -0.724117767  0.3845059020 -1.1215802492
## [100,]  0.6009151124 -0.035508139 -0.1301554621 -0.1594497275
## [101,] -0.0017887133 -0.188474718  0.0005162167 -0.4173992574
## [102,] -0.2065423206 -0.090622894  0.1045334324  0.3125803587
## [103,] -0.4953081412 -0.198615060  0.5375937084  0.2047964522
## [104,]  0.8686428100 -0.093770859 -0.4773834996 -0.5141971675
## [105,] -0.1548423582  0.058531955  0.0446714809  0.0792353509
## [106,]  0.1318303043  0.037461065  0.4258940341 -0.5815966071
## [107,]  0.1564602678 -0.292990155  0.2050456375  0.2709016999
## [108,]  0.1287033758 -0.078424785 -0.4344151819  0.3056587758
## [109,] -0.0180890292  0.267698463  0.1023578502 -0.3150970929
## [110,] -0.2396821206 -0.307689778 -0.0699041488  0.1148243302
## [111,]  0.0713294985  0.167676166  0.4831425613  0.3885721928
## [112,] -0.3063869904  0.520204354  0.0765713670  0.6721078299
## [113,] -0.2882665833 -0.092850152  0.1895246446 -0.2214078341
## [114,]  0.1001758633 -0.360988377 -0.0636838627 -0.1289911375
## [115,] -0.0108778023 -0.025341714  0.1467266897  0.5493639263
## [116,]  0.7106153360 -0.580261933  0.1418017754 -0.3225882103
## [117,]  0.0803582529 -0.148644666 -0.2434786147  0.0120862192
## [118,] -0.3434085839  0.168098335 -0.2329438011  0.1719473665
## [119,] -0.1835033096 -0.104590792  0.0416161742 -0.4010998024
## [120,]  0.0587627705  0.235668343 -0.2196746434 -0.0606561810
## [121,] -0.4248134747  0.424071090  0.2221360432  0.3460519534
## [122,]  0.7346517342 -0.190767021  0.5383947222 -0.3306323578
## [123,]  0.5986743960  0.195866244  0.7155286706 -0.1437084258
## [124,]  0.0966111080 -0.114117289  0.1807983282  0.0813367239
## [125,]  0.2514274285 -0.713947740 -0.6311619599  0.8903277070
## [126,] -0.4783323716 -0.138145257  0.1445270256 -0.0178799576
## [127,] -0.0464856928 -0.255405449 -0.1041353493 -0.6475405731
## [128,] -0.0864144711 -0.132693368 -0.4329269704  0.0397529724
## [129,]  0.1123182610 -0.063808707  0.4299455635  0.2305618715
## [130,] -0.3255923965  0.044612379 -0.1974110369 -0.2730104553
## [131,] -0.1896524529  0.159783600  0.8850629603 -0.1300808223
## [132,] -0.0744540819 -0.062603410  0.2501150995  0.3224576986
## [133,] -0.1864585990  0.135573834  0.2583707289  0.2411101270
## [134,]  0.0312512889 -0.058770819 -0.1493635832  0.2396250410
## [135,]  0.2236790945  0.099295026  0.2797190025 -0.1014961284
## [136,] -0.2904532284  0.062227032 -0.2043717144  0.0076293619
## [137,]  0.2980513984 -0.018604443  0.1224390359 -0.2684118806
## [138,] -0.2904532284  0.062227032 -0.2043717144  0.0076293619
## [139,] -0.1377132533  0.244984668 -0.5463097769 -0.2475011358
## [140,]  0.5698917238 -0.613765822  0.1984662459 -0.6695779330
## [141,]  0.1001758633 -0.360988377 -0.0636838627 -0.1289911375
## [142,] -0.0257522848  0.214482653 -0.0754882177 -0.1240230785
## [143,] -0.5954511794  0.102329594  0.3393147403 -0.2408578443
## [144,]  0.1654368504  0.331788048 -0.0383178619  0.3312268427
## [145,] -0.3691340054  0.766670528  0.5874918164  0.0177326008
## [146,] -0.1819963022  0.414432397 -0.4646285665 -0.5152546619
## [147,]  0.1325902764 -0.240058029  0.2995559584  0.2242427719
## [148,] -0.0768997539 -0.221030285  0.0782577495 -0.0481482200
## [149,]  0.5842967725 -0.813913201  0.4320125532  0.2136991586
## [150,]  0.0488112280  0.240775077 -0.1081968166 -0.2780876374
## [151,] -0.0153498605 -0.083615275 -0.2900232311 -0.0421413334
## [152,]  0.4398691629 -0.736939232 -0.2621339450 -0.0716172945
## [153,]  0.2291151179 -0.267829687 -0.1429153910 -0.0356391152
## [154,] -0.3255923965  0.044612379 -0.1974110369 -0.2730104553
## [155,]  0.3945884893 -0.104922376 -0.0857987995 -1.5192361371
## [156,] -0.6296293402  0.605745730 -0.8453862587 -0.4996153657
## [157,]  0.0278771609 -0.164903676  0.1414614379 -0.0591332258
## [158,] -0.4783323716 -0.138145257  0.1445270256 -0.0178799576
## [159,]  0.6270866900  0.369142424 -0.0366457738 -0.4556268142
## [160,]  0.1046692425  0.177152845 -0.0695756362  0.0807541853
## [161,] -0.1763837130 -0.093748054  0.4727839032 -0.9473651420
## [162,] -0.4988823674  0.426874219  0.0846370574 -0.2673908202
## [163,] -0.4783323716 -0.138145257  0.1445270256 -0.0178799576
## [164,]  0.2658228099  0.257610501  0.0906778503  0.0832186941
## [165,] -0.2904532284  0.062227032 -0.2043717144  0.0076293619
## [166,] -0.2032290037 -0.283987564  0.0588755089 -0.0676506530
## [167,]  0.0966111080 -0.114117289  0.1807983282  0.0813367239
## [168,]  0.9112016944  0.100772835 -0.1915258990 -0.6692947810
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## [170,] -0.4147922267  0.046324263 -0.3901307067 -0.2804660913
## [171,] -0.3255923965  0.044612379 -0.1974110369 -0.2730104553
## [172,]  0.2581927951  0.717343929  0.3988032418  0.0564581082
## [173,] -0.0761610172 -0.070575249  0.1577961833 -0.2504835052
## [174,]  0.1072393312  0.392179851  0.4184201926 -0.0026305025
## [175,]  0.1325902764 -0.240058029  0.2995559584  0.2242427719
## [176,] -0.0153498605 -0.083615275 -0.2900232311 -0.0421413334
## [177,] -0.0296565433 -0.227837832 -0.0465038657 -0.0705726322
## [178,]  0.1143622727  0.284171570  0.7189310299 -0.0355330345
## [179,]  0.1661293327  0.295202890 -0.7263048271  0.5859132482
## [180,]  0.0102268836  0.088541914  0.0432694126  0.0188829695
## [181,]  0.3675446821 -0.359614370 -0.1291642606  0.8700780834
## [182,]  0.0549586712  0.241735966 -0.8422702845  0.1195365582
## [183,]  0.4389388860  0.702363029 -0.1774979024 -0.2606231314
## [184,] -0.2272291958 -0.331525001  0.1630383190  0.1536911009
## [185,] -0.4340279400  0.119315918  0.1208630748  0.3433586847
## [186,]  0.3868160088 -0.051641724 -0.5441012912 -0.2721735466
## [187,]  0.1944501592 -0.052084200  0.1593490009  0.0591600307
## [188,] -0.2032290037 -0.283987564  0.0588755089 -0.0676506530
## [189,] -0.4783323716 -0.138145257  0.1445270256 -0.0178799576
## [190,] -0.0761610172 -0.070575249  0.1577961833 -0.2504835052
## [191,] -0.5048970330 -0.276223619  0.1344117642 -0.5872586738
## [192,]  0.0117660765 -0.061140933 -0.1446777262 -0.1317918640
## [193,] -0.0153498605 -0.083615275 -0.2900232311 -0.0421413334
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## [681,]  0.4052218492
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## [683,]  0.1499127847
BreastCancer_clean$NewScore <- mjca1$rowcoord[, 1] #dim1 

head(BreastCancer_clean)
##        Id Cl.thickness Cell.size Cell.shape Marg.adhesion Epith.c.size
## 1 1000025            5         1          1             1            2
## 2 1002945            5         4          4             5            7
## 3 1015425            3         1          1             1            2
## 4 1016277            6         8          8             1            3
## 5 1017023            4         1          1             3            2
## 6 1017122            8        10         10             8            7
##   Bare.nuclei Bl.cromatin Normal.nucleoli Mitoses     Class   NewScore
## 1           1           3               1       1    benign  0.8843631
## 2          10           3               2       1    benign -0.9870290
## 3           2           3               1       1    benign  0.8474861
## 4           4           3               7       1    benign -0.9310026
## 5           1           3               1       1    benign  0.7956928
## 6          10           9               7       1 malignant -2.1731476

Now, let’s check whether the factors are independent

(Chi-Sq scores for significance of correlation - smaller the stronger is the fit/relationship and Cramer’s V for strength of it - the opposite. )

features1 <- cbind( BreastCancer_clean$Cell.size, BreastCancer_clean$Cell.shape )
chsq1<- chisq.test(features1)
## Warning in chisq.test(features1): Chi-squared approximation may be
## incorrect
#X-squared = 131.71, df = 682, p-value = 1. There is no relation

library(lsr)
crv1<- cramersV(features1)
## Warning in chisq.test(...): Chi-squared approximation may be incorrect
#0.1740442 : There is weak relations between Cell size and Cell shape


features2 <- cbind( BreastCancer_clean$Cell.size, BreastCancer_clean$Cl.thickness )
chsq2<- chisq.test(features2)
## Warning in chisq.test(features2): Chi-squared approximation may be
## incorrect
#X-squared = 538.53, df = 682, p-value = 1. There is no relation

crv2<- cramersV(features2)
## Warning in chisq.test(...): Chi-squared approximation may be incorrect
#0.3222482 : There is not strong relations between Cell size and Cell thickness


features3 <- cbind( BreastCancer_clean$Cell.size, BreastCancer_clean$Margin.adhesion )
chsq3<- chisq.test(features3)
## Warning in chisq.test(features3): Chi-squared approximation may be
## incorrect
#X-squared = 2033.6, df = 682, p-value < 2.2e-16. There might be some relation

crv3<- cramersV(features3)
## Warning in chisq.test(...): Chi-squared approximation may be incorrect
#0.03722366 : However, Cramer's scores tell us that that the relationship is not strong


features4 <- cbind( BreastCancer_clean$Cell.size, BreastCancer_clean$ Epith.c.size)
chsq4<- chisq.test(features4)
## Warning in chisq.test(features4): Chi-squared approximation may be
## incorrect
#X-squared = 329.42, df = 682, p-value = 1. No relation

crv4<- cramersV(features4)
## Warning in chisq.test(...): Chi-squared approximation may be incorrect
#0.2748433 : The relationship is not strong

# Creating table of statistical scores for features correaltion:
library(broom)
library(purrr)

ChiSq_table <- map_df(list(chsq1, chsq2, chsq3, chsq4), tidy)
#to get just the columns I need:
ChiSq_table[c("statistic", "p.value", "method")] 
## # A tibble: 4 x 3
##   statistic    p.value method                                  
##       <dbl>      <dbl> <chr>                                   
## 1      132.  1.00e+  0 Pearson's Chi-squared test              
## 2      539. 10.00e-  1 Pearson's Chi-squared test              
## 3     2034.  2.18e-134 Chi-squared test for given probabilities
## 4      329.  1.00e+  0 Pearson's Chi-squared test
# statistic is X-sq score. method is the df

With chi-sq and Cramer’s V scores confirmed that the features we have on hands are not collinear. We can proceed for LR.

Saving MCA rowscore as a new feature in dataset. Splitting data. Building LR model on the train, testing it. Plotting ROC curve and interpreting AUC score.

# I will then use "NewScore" as one of the predictors for the model instead of original 9 variables. 
#data(BreastCancer_clean)

#split into training and test set 
train_size <- floor(0.75 * nrow(BreastCancer_clean))
set.seed(324)
train_Index <- sample(seq_len(nrow(BreastCancer_clean)), size = train_size)
train_classifier <- BreastCancer_clean[train_Index,]
test_classifier <- BreastCancer_clean[-train_Index,]


dim(train_classifier) #[512:12]
## [1] 512  12
dim(test_classifier) #[171:12]
## [1] 171  12
#Model:
MCA_glm <- glm(Class ~ NewScore, data = train_classifier, family = binomial)
summary(MCA_glm)
## 
## Call:
## glm(formula = Class ~ NewScore, family = binomial, data = train_classifier)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.96613  -0.09627  -0.06609   0.07182   2.02015  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -1.7990     0.3614  -4.978 6.44e-07 ***
## NewScore     -4.0326     0.4462  -9.039  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 668.731  on 511  degrees of freedom
## Residual deviance:  86.263  on 510  degrees of freedom
## AIC: 90.263
## 
## Number of Fisher Scoring iterations: 8
#Intercept: -1.7990  NewScore -4.0326
#Trying more elaborate LR with Control:

ctl_BRCA <- trainControl(method = "repeatedcv", repeats = 10,classProbs = T,
                     savePredictions = T)
#summary(ctrl_BRCA)

MCA_train <- train(Class ~ NewScore, data = train_classifier, method = "glm", family= "binomial", trControl = ctl_BRCA)

summary(MCA_train)
## 
## Call:
## NULL
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.96613  -0.09627  -0.06609   0.07182   2.02015  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -1.7990     0.3614  -4.978 6.44e-07 ***
## NewScore     -4.0326     0.4462  -9.039  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 668.731  on 511  degrees of freedom
## Residual deviance:  86.263  on 510  degrees of freedom
## AIC: 90.263
## 
## Number of Fisher Scoring iterations: 8
#Intercept: -1.7990  NewScore -4.0326
# SAME RESULT!!!!

Testing the model and evaluating its accuracy (ROC and AUC):

#Test set:
library(pROC)

MCA_glm_predicted <- predict(MCA_glm, newdata=test_classifier, type="response") 

# Note: feed ROC not with vectors (predicted and actual) as here:
# roccurve <- roc(data=test_classifier, MCA_glm$Class, predicted_classification) 

# But with a single matrix instead:
predicted_data <- data.frame(prob = MCA_glm_predicted,
                             class = test_classifier$Class)

roccurve <- roc(data = predicted_data, response = "class", "prob" ) 
## Setting levels: control = benign, case = malignant
## Setting direction: controls < cases
# roc() calculates confusion matrix for test set and outputs AUC score
#another option is to use confusionMatrix()

plot(roccurve)  # evaluation of a model performance on test to evaluate how well NewScore can predict malignancy state(=1) of breast cancer

#auc(roccurve)

Both the ROC curve and the high AUC value (Area Under the Curve) indicate that the model is good in predicting no DM (i.e. true positives).

Vizualizing the results:

! Checking if log odds and independent variables are linearly correlated

# To convert from a probability to odds, divide the probability by one minus that probability. So if the probability is 10% or 0.10 , then the odds are 0.1/0.9 or ‘10%/90%’ or 1 out of 9 
prediction_odds <- MCA_glm_predicted / (1 - MCA_glm_predicted)
log_prediction_odds <- log(prediction_odds)

#log_odds and indep.variable X must be linearly correlated:
cor.test(log_prediction_odds, test_classifier$NewScore)
## 
##  Pearson's product-moment correlation
## 
## data:  log_prediction_odds and test_classifier$NewScore
## t = -Inf, df = 169, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -1 -1
## sample estimates:
## cor 
##  -1
plot(log_prediction_odds, test_classifier$NewScore)

# Yes, they are correlated!

plotting predicted vs actual obs.

#prepare df to plot = predicted + actual labels:
df_to_plot_results <- data.frame(Class_pred = MCA_glm_predicted, MCA_selected_features  = test_classifier$NewScore,
                                 classes = test_classifier$Class) 

#Stash away actual labels as logical vector:
df_to_plot_results$Class <- as.numeric(test_classifier$Class) -1

# Plotting predictions and actual observations: 
ggplot(data = df_to_plot_results) +
  geom_point(aes(x=MCA_selected_features, y = Class)) + 
  geom_line(aes(x = MCA_selected_features, y = Class_pred, col = "Blue")) +
  geom_line(aes(x = MCA_selected_features, y = 1-Class_pred,  col = "Red"))+ 
#(x and y whatever you name it, input data is the only thing to worry here.)
  ggtitle("Probabilities for classifying whether benign or malignant")

CONCLUSION on MCA and Logistic Regression:

" > roccurve

Call: roc.data.frame(data = predicted_data, response = “class”, predictor = “prob”)

Data: prob in 116 controls (class benign) < 55 cases (class malignant). Area under the curve: 0.9892 "

Conclusion 1: Selected single Dim1 was a successful choice to select features for Logistic Regression. The model performed with TP of 98% on validation set, which is pretty high.

Conclusion 2. What could be done more: 1. Evaluation of the model right after building it.

The other ways to go about the task: 1. as.numeric(data) -> PCA -> Logistic Regression 2. kNN 3. data[-Class] -> k-Means with k=2 and also choosing k through silhouette analysis. It will be interesting to compare these results. Will do it when bit later.

The end. Thank you!

===================

Optional- exploring other packages to perform exactly same steps as before:

MCA performed with FactoMineR::MCA

library(FactoMineR)
library(factoextra)

res.mca <- MCA(BreastCancer_MCA, graph = T)

print(res.mca)
## **Results of the Multiple Correspondence Analysis (MCA)**
## The analysis was performed on 683 individuals, described by 9 variables
## *The results are available in the following objects:
## 
##    name              description                       
## 1  "$eig"            "eigenvalues"                     
## 2  "$var"            "results for the variables"       
## 3  "$var$coord"      "coord. of the categories"        
## 4  "$var$cos2"       "cos2 for the categories"         
## 5  "$var$contrib"    "contributions of the categories" 
## 6  "$var$v.test"     "v-test for the categories"       
## 7  "$ind"            "results for the individuals"     
## 8  "$ind$coord"      "coord. for the individuals"      
## 9  "$ind$cos2"       "cos2 for the individuals"        
## 10 "$ind$contrib"    "contributions of the individuals"
## 11 "$call"           "intermediate results"            
## 12 "$call$marge.col" "weights of columns"              
## 13 "$call$marge.li"  "weights of rows"
#Other way:
#mca_output <- MCA(BreastCancer_MCA, ncp=2, ind.sup = NULL, quanti.sup = NULL, quali.sup = NULL, excl=NULL, graph = TRUE, level.ventil = 0, axes = c(1,2), row.w = NULL, method="Indicator", na.method="NA", tab.disj=NULL)
 
#summary(mca_output)

Summary of MCA outputs with FactoMineR::summary.MCA().

Arguments: object: an object of class MCA nb.dec: number of decimal printed nbelements: number of row/column variables to be written. To have all the elements, use nbelements = Inf. ncp: Number of dimensions to be printed

Print the summary of the MCA for the dimensions 1 and 2:

MCA_summary<- summary(res.mca, nb.dec = 2, ncp = 2)
## 
## Call:
## MCA(X = BreastCancer_MCA, graph = T) 
## 
## 
## Eigenvalues
##                       Dim.1  Dim.2  Dim.3  Dim.4  Dim.5  Dim.6  Dim.7
## Variance               0.72   0.31   0.23   0.21   0.20   0.20   0.19
## % of var.              8.11   3.51   2.62   2.37   2.30   2.24   2.16
## Cumulative % of var.   8.11  11.62  14.24  16.60  18.90  21.15  23.30
##                       Dim.8  Dim.9 Dim.10 Dim.11 Dim.12 Dim.13 Dim.14
## Variance               0.19   0.18   0.18   0.17   0.17   0.17   0.16
## % of var.              2.10   2.07   2.02   1.95   1.90   1.88   1.81
## Cumulative % of var.  25.41  27.48  29.49  31.44  33.34  35.22  37.03
##                      Dim.15 Dim.16 Dim.17 Dim.18 Dim.19 Dim.20 Dim.21
## Variance               0.16   0.16   0.15   0.15   0.14   0.14   0.14
## % of var.              1.76   1.75   1.68   1.66   1.63   1.61   1.58
## Cumulative % of var.  38.79  40.54  42.23  43.89  45.52  47.12  48.71
##                      Dim.22 Dim.23 Dim.24 Dim.25 Dim.26 Dim.27 Dim.28
## Variance               0.14   0.13   0.13   0.13   0.13   0.12   0.12
## % of var.              1.56   1.51   1.48   1.46   1.41   1.39   1.37
## Cumulative % of var.  50.27  51.78  53.26  54.72  56.13  57.51  58.89
##                      Dim.29 Dim.30 Dim.31 Dim.32 Dim.33 Dim.34 Dim.35
## Variance               0.12   0.12   0.12   0.12   0.11   0.11   0.11
## % of var.              1.35   1.33   1.32   1.30   1.29   1.23   1.21
## Cumulative % of var.  60.23  61.56  62.88  64.19  65.48  66.70  67.91
##                      Dim.36 Dim.37 Dim.38 Dim.39 Dim.40 Dim.41 Dim.42
## Variance               0.11   0.11   0.10   0.10   0.10   0.10   0.09
## % of var.              1.20   1.19   1.16   1.12   1.11   1.08   1.06
## Cumulative % of var.  69.11  70.30  71.46  72.57  73.68  74.76  75.82
##                      Dim.43 Dim.44 Dim.45 Dim.46 Dim.47 Dim.48 Dim.49
## Variance               0.09   0.09   0.09   0.09   0.09   0.09   0.08
## % of var.              1.05   1.03   1.02   1.00   0.98   0.96   0.94
## Cumulative % of var.  76.87  77.90  78.92  79.91  80.89  81.85  82.80
##                      Dim.50 Dim.51 Dim.52 Dim.53 Dim.54 Dim.55 Dim.56
## Variance               0.08   0.08   0.08   0.08   0.07   0.07   0.07
## % of var.              0.92   0.90   0.88   0.84   0.79   0.77   0.76
## Cumulative % of var.  83.71  84.62  85.50  86.34  87.13  87.90  88.66
##                      Dim.57 Dim.58 Dim.59 Dim.60 Dim.61 Dim.62 Dim.63
## Variance               0.07   0.06   0.06   0.06   0.06   0.06   0.05
## % of var.              0.74   0.73   0.69   0.68   0.66   0.64   0.62
## Cumulative % of var.  89.41  90.14  90.83  91.51  92.17  92.81  93.43
##                      Dim.64 Dim.65 Dim.66 Dim.67 Dim.68 Dim.69 Dim.70
## Variance               0.05   0.05   0.05   0.05   0.04   0.04   0.04
## % of var.              0.60   0.58   0.56   0.53   0.50   0.49   0.46
## Cumulative % of var.  94.02  94.60  95.16  95.69  96.19  96.68  97.13
##                      Dim.71 Dim.72 Dim.73 Dim.74 Dim.75 Dim.76 Dim.77
## Variance               0.04   0.04   0.03   0.03   0.03   0.02   0.02
## % of var.              0.44   0.41   0.39   0.33   0.31   0.27   0.24
## Cumulative % of var.  97.58  97.99  98.38  98.71  99.01  99.28  99.52
##                      Dim.78 Dim.79 Dim.80
## Variance               0.02   0.02   0.01
## % of var.              0.21   0.19   0.07
## Cumulative % of var.  99.73  99.93 100.00
## 
## Individuals (the 10 first)
##                   Dim.1   ctr  cos2   Dim.2   ctr  cos2  
## 1               | -0.61  0.07  0.27 | -0.09  0.00  0.01 |
## 2               |  0.68  0.09  0.03 |  0.57  0.15  0.02 |
## 3               | -0.58  0.07  0.09 | -0.04  0.00  0.00 |
## 4               |  0.64  0.08  0.02 |  0.26  0.03  0.00 |
## 5               | -0.55  0.06  0.10 |  0.00  0.00  0.00 |
## 6               |  1.49  0.45  0.09 | -1.08  0.54  0.05 |
## 7               | -0.45  0.04  0.12 | -0.10  0.00  0.01 |
## 8               | -0.65  0.08  0.12 | -0.01  0.00  0.00 |
## 9               | -0.58  0.07  0.02 | -0.43  0.09  0.01 |
## 10              | -0.68  0.09  0.14 | -0.02  0.00  0.00 |
## 
## Categories (the 10 first)
##                    Dim.1    ctr   cos2 v.test    Dim.2    ctr   cos2
## Cl.thickness_1  |  -0.76   1.83   0.15 -10.08 |  -0.19   0.27   0.01
## Cl.thickness_2  |  -0.65   0.47   0.03  -4.76 |  -0.05   0.01   0.00
## Cl.thickness_3  |  -0.50   0.58   0.04  -5.48 |   0.12   0.08   0.00
## Cl.thickness_4  |  -0.45   0.35   0.03  -4.21 |   0.06   0.01   0.00
## Cl.thickness_5  |   0.05   0.01   0.00   0.67 |   0.00   0.00   0.00
## Cl.thickness_6  |   0.69   0.36   0.02   4.09 |  -0.21   0.07   0.00
## Cl.thickness_7  |   1.28   0.85   0.06   6.22 |   1.44   2.50   0.07
## Cl.thickness_8  |   1.29   1.65   0.11   8.84 |   0.05   0.01   0.00
## Cl.thickness_9  |   1.35   0.57   0.04   5.09 |  -0.83   0.50   0.01
## Cl.thickness_10 |   1.31   2.68   0.19  11.49 |  -0.06   0.01   0.00
##                 v.test  
## Cl.thickness_1   -2.57 |
## Cl.thickness_2   -0.38 |
## Cl.thickness_3    1.30 |
## Cl.thickness_4    0.56 |
## Cl.thickness_5   -0.01 |
## Cl.thickness_6   -1.21 |
## Cl.thickness_7    7.04 |
## Cl.thickness_8    0.38 |
## Cl.thickness_9   -3.14 |
## Cl.thickness_10  -0.55 |
## 
## Categorical variables (eta2)
##                   Dim.1 Dim.2  
## Cl.thickness    |  0.61  0.10 |
## Cell.size       |  0.90  0.64 |
## Cell.shape      |  0.86  0.62 |
## Marg.adhesion   |  0.71  0.26 |
## Epith.c.size    |  0.79  0.34 |
## Bare.nuclei     |  0.75  0.07 |
## Bl.cromatin     |  0.76  0.28 |
## Normal.nucleoli |  0.73  0.26 |
## Mitoses         |  0.38  0.23 |
#loading_score<- MCA_summary$CategoricalVariables_(eta2)[, 1]

Note: For exporting the summary to a file, use the code: summary(res.mca, file =“myfile.txt”) For displaying the summary of more than 10 elements, use the argument nbelements in the function summary()

####Interpretation of MCA outputs (http://www.sthda.com/english/articles/31-principal-component-methods-in-r-practical-guide/113-ca-correspondence-analysis-in-r-essentials/)

Eigenvalues/variances and screeplot The proportion of variances retained by the different dimensions (axes) can be extracted using the function get_eigenvalue() [in factoextra] as follow :

eigenvalues <- get_eigenvalue(res.mca)
head(round(eigenvalues, 2))
##       eigenvalue variance.percent cumulative.variance.percent
## Dim.1       0.72             8.11                        8.11
## Dim.2       0.31             3.51                       11.62
## Dim.3       0.23             2.62                       14.24
## Dim.4       0.21             2.37                       16.60
## Dim.5       0.20             2.30                       18.90
## Dim.6       0.20             2.24                       21.15

SCREEPLOT factoextra::fviz_screeplot() (the percentages of inertia explained by the MCA dimensions):

fviz_screeplot(res.mca)

# It is possible that the two dimensions we chose are sufficient to explain data variance

(more: http://www.sthda.com/english/wiki/eigenvalues-quick-data-visualization-with-factoextra-r-software-and-data-mining)

MCA SCATTER plot: (Biplot of observ. and variable factors linked together) FactoMineR::plot.MCA()

# a simple format:
plot.MCA(res.mca)#, axes = c(1,2), choix=c("ind", "var"))

#plot(res.mca)

Note: x : An object of class MCA axes : A numeric vector of length 2 specifying the component to plot choix : The graph to be plotted. Possible values are “ind” for the individuals and “var” for the variables

same MCA BIPLOT with different package: factoextra::fviz_mca_biplot():

fviz_mca_biplot(res.mca)

# Change the theme
#fviz_mca_biplot(res.mca) +
#  theme_minimal()

CONCLUSION ON BIPLOT: a global pattern within the data: rows (individuals) are represented by blue points and columns (variable categories) by red triangles. The distance between any row points or column points gives a measure of their similarity (or dissimilarity). Row points with similar profile are closed on the factor map. The same holds true for column points.

VARIABLE CATEGORIES

factoextra::get_mca_var() used to extract the results for variable categories. vs ca::mjca$rowcoord or $rowpcoord Output is a list containing the coordinates, the cos2 and the contribution of variable categories.

var <- get_mca_var(res.mca)
var
## Multiple Correspondence Analysis Results for variables
##  ===================================================
##   Name       Description                  
## 1 "$coord"   "Coordinates for categories" 
## 2 "$cos2"    "Cos2 for categories"        
## 3 "$contrib" "contributions of categories"

Variables can be visualized:

plot(res.mca, choix = "var")

Cell.size and Cell.shape are correlated to both dimensions.

There is no given variable strongly correlated only to dim.2 alone…

The coordinates are the squared correlations between variables and the dimensions.

However, we need to investigate more.

Extracting Coordinates of variable categories to be used a s new predictors in model:

head(round(var$coord, 2))
##                Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
## Cl.thickness_1 -0.76 -0.19  0.17 -0.05 -0.13
## Cl.thickness_2 -0.65 -0.05  0.12  0.02 -0.33
## Cl.thickness_3 -0.50  0.12  0.01 -0.06 -0.01
## Cl.thickness_4 -0.45  0.06  0.15  0.01  0.31
## Cl.thickness_5  0.05  0.00 -0.34 -0.04 -0.02
## Cl.thickness_6  0.69 -0.21  0.36  0.52 -0.42

visualize only variable categories: (factoextra::fviz_mca_var())

# Default plot
#fviz_mca_var(res.mca)
fviz_mca_var(res.mca, col.var="black", shape.var = 70)

MEASURING CONTRIBUTION of variable categories (%) to the dimensions

head(round(var$contrib,2))
##                Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
## Cl.thickness_1  1.83  0.27  0.28  0.02  0.18
## Cl.thickness_2  0.47  0.01  0.05  0.00  0.44
## Cl.thickness_3  0.58  0.08  0.00  0.03  0.00
## Cl.thickness_4  0.35  0.01  0.12  0.00  0.60
## Cl.thickness_5  0.01  0.00  1.03  0.02  0.01
## Cl.thickness_6  0.36  0.07  0.30  0.70  0.47

The variable categories with the larger value, contribute the most to the definition of the dimensions.

The different categories in the table are:

categories <- rownames(var$coord)
length(categories)
## [1] 89

there are initial 89 categories in our data.

HIGHLIGHTs on MOST CONTR-G VAR-S for each dimension:

#install.packages("corrplot")
library("corrplot")
## corrplot 0.84 loaded
head(corrplot(var$contrib, is.corr = FALSE))

##                       Dim 1        Dim 2        Dim 3        Dim 4
## Cl.thickness_1 0.1378117042 0.0207148626 2.127337e-02 1.869176e-03
## Cl.thickness_2 0.0357631253 0.0005350944 3.990319e-03 1.017286e-04
## Cl.thickness_3 0.0434030398 0.0056765779 2.574009e-05 2.167341e-03
## Cl.thickness_4 0.0267095347 0.0011044249 9.370846e-03 2.942556e-05
## Cl.thickness_5 0.0006165451 0.0000000000 7.779277e-02 1.475081e-03
## Cl.thickness_6 0.0270696589 0.0055143641 2.237007e-02 5.249592e-02
##                       Dim 5
## Cl.thickness_1 0.0133554483
## Cl.thickness_2 0.0329050392
## Cl.thickness_3 0.0001171677
## Cl.thickness_4 0.0453055616
## Cl.thickness_5 0.0003907063
## Cl.thickness_6 0.0353811895

Bar plot of variable contributions: (factoextra::fviz_contrib())

# Contributions of variables on Dim.1
fviz_contrib(res.mca, choice = "var", axes = 1)

If the contribution of variable categories were uniform, the expected value would be 1/number_of_categories = 1/22 = 4.5%.

The red dashed line on the graph above indicates the expected average contribution. For a given dimension, any category with a contribution larger than this threshold could be considered as important in contributing to that dimension.

It can be seen that the categories … are the most important in the definition of the first dimension.

# Contributions of rows on Dim.2
fviz_contrib(res.mca, choice = "var", axes = 2)

Correlation between variables and principal dimensions

# Total contribution on Dim.1 and Dim.2
fviz_contrib(res.mca, choice = "var", axes = 1:2)

The total contribution of a category, on explaining the variations retained by Dim.1 and Dim.2, is calculated as follow : (C1 * Eig1) + (C2 * Eig2).

C1 and C2 are the contributions of the category to dimensions 1 and 2, respectively. Eig1 and Eig2 are the eigenvalues of dimensions 1 and 2, respectively.

The expected average contribution of a category for Dim.1 and Dim.2 is : (4.5 * Eig1) + (4.5 * Eig2) = (4.50.34) + (4.50.13) = 2.12%

If data contains many categories, the TOP CONTR-ING categories

fviz_contrib(res.mca, choice = "var", axes = 1, top = 10)

fviz_contrib(res.mca, choice = "var", axes = 1:2, top = 10)

#####CONCLUSION:

According to the categories contribution barplot, Cell.size, Cell.shape, Normal.nucleoli, Bare.nuclei, Marg.adhesion and Epith.c.size_2 will be included in Logistic Regression with their factors.

#Optional:

#A second option is to draw a SCATTERPLOT of categories and to highlight categories according to the amount of their contributions.

# Control category point colors using their contribution
# Possible values for the argument col.row are :
  # "cos2", "contrib", "coord", "x", "y"
fviz_mca_var(res.mca, col.var = "contrib")+
scale_color_gradient2(low="white", mid="blue", 
                      high="red", midpoint=2)+theme_minimal()

# Optional:

#TRANSPARENCY BY CONTRIB.
#fviz_mca_var(res.mca, alpha.var="contrib")+
  #theme_minimal()
#Optional:

# TOP 10 contributing categories
fviz_mca_var(res.mca, select.var=list(contrib=10))

#Optional:
  
#Change individual colors by groups using the levels of the variable sick. The argument with factors and subfactors desired ='habillage':
#fviz_mca_ind(res.mca, label = "none", Cell.size=10)

#Add ellipses of point concentrations : the argument habillage is used to specify the factor variable for coloring the observations by groups.

#fviz_mca_ind(res.mca, label="none", Cell.shape=10,
 #            addEllipses = TRUE, ellipse.level = 0.95)
#Optional:

#top 5 contributing individuals and variable categories
fviz_mca_biplot(res.mca, select.ind = list(contrib = 5), 
               select.var = list(contrib = 5)) +
  theme_minimal()

#Supplementary individuals/variable categories are not shown because they don’t contribute to the construction of the axes.
# Extracting coordinates for MCA individuals/observations:
coord = fviz_mca_ind(res.mca, select.ind = list(contrib = 10))+ 
  theme_minimal()
coord


References: https://sebastianraschka.com/Articles/2014_python_lda.html

https://towardsdatascience.com/building-a-multiple-linear-regression-model-and-assumptions-of-linear-regression-a-z-9769a6a0de42

http://www.statisticssolutions.com/wp-content/uploads/wp-post-to-pdf-enhanced-cache/1/assumptions-of-logistic-regression.pdf

https://machinelearningmastery.com/linear-discriminant-analysis-for-machine-learning/ , https://sebastianraschka.com/Articles/2014_python_lda.html

Other cool sites: https://www.countbayesie.com/blog/2019/6/12/logistic-regression-from-bayes-theorem https://colah.github.io/posts/2014-03-NN-Manifolds-Topology/